dissertation: spectrum sensing and interference mitigation in … · 4 utilization of such...

Spectrum Sensing and Interference Mitigationin Cognitive Radio Networks

Von der Fakultät für Elektrotechnik und Informationstechnikder Rheinisch–Westfälischen Technischen Hochschule Aachen

zur Erlangung des akademischen Gradeseiner Doktorin der Ingenieurwissenschaften

genehmigte Dissertation

vorgelegt vonXitao Gong, M.Sc.

aus Nanjing, China

Berichter: Universitätsprofessor Dr.-Ing. Gerd Ascheid

Universitätsprofessor Dr. Petri Mähönen

Tag der mündlichen Prüfung: 21.02.2014

Diese Dissertation ist auf den Internetseitender Hochschulbibliothek online verfügbar.

Abstract

One concept to increase spectral efficiency is dynamic spectrum access (DSA). The sec-ondary users coexist with the primary users in the same radio-frequency spectrum.Such shared usage demands for an interference avoidance or interference mitigationat the secondary users. This is especially challenging due to the limited cooperationbetween the primary users (PUs) and the secondary users (SUs). Motivated by thisfact, the focus of this thesis is a comprehensive study on interference managementstrategies and the characterization of the achievable performance of secondary sys-tems.

Spectrum sensing aims at detecting the presence or absence of the PUs. The mainchallenge encountered is the high requirement on sensitivity, reliability, and agility,especially in case of incomplete knowledge of the transmission channels. Therefore,the SUs need to efficiently utilize the limited a priori knowledge related to the pri-mary transmission to improve the sensing performance. In this thesis, the generalizedlikelihood ratio test framework is applied to cooperative sensing problems with an un-known structure of the primary signal space and unknown noise variances at the SUs.The efficiency of the resulting spectrum sensing algorithms is demonstrated as wellas the effectiveness in countering the “hidden primary user” problem.

Based on limited knowledge related to the primary transmission, the SUs’ transcei-ver strategies are optimized in order to achieve the tradeoff between improved sec-ondary network throughput and, most critically, constrain the performance loss ofthe primary transmission. For a single-antenna spectrum sharing system, the powerallocation strategies are investigated for the SUs subject to different quality of serviceconstraints on the primary link. Not only optimal and low-complexity near-optimalpower allocation strategies are developed, but also the achievable performance of thesystem is approximately evaluated in closed form. Additionally, for multi-antennaspectrum sharing networks, efficient transceiver optimization strategies are devel-oped under the consideration of imperfect channel state information. The robust-ness, optimality, and convergence behavior of the different proposed algorithms arequantitatively verified and compared.

The essential “cognitive” property of DSA in cognitive radio networks consists oftwo aspects: the acquirement of the useful information from the environment and the

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utilization of such information to improve the spectrum efficiency. This is demon-strated with the study of a hybrid paradigm, in which the SU exploits the spectrumsensing and location information to adapt the transmit power level. Compared to thestandard paradigms, e.g., opportunistic transmission and spectrum sharing withoutsensing, the proposed strategies in the hybrid paradigm achieve better performance.

Kurzfassung

Ein Konzept zur Erhöhung der spektralen Effizienz ist der dynamische Zugang zumSpektrum (Dynamic Spectrum Access, DSA). Sekundäre Nutzer koexistieren dabeimit den primären Nutzern eines Frequenzbands. Eine solche gemeinsame Nutzungsetzt Interferenzvermeidung bzw. Interferenzminimierung durch die sekundären Be-nutzer voraus. Dies ist insbesondere durch die begrenzte Kooperation zwischen denprimären Benutzern (Primary Users, PU) und den sekundären Nutzern (SecondaryUsers, SU) eine Herausforderung. Schwerpunkt dieser Arbeit ist eine umfassende Un-tersuchung möglicher Interferenz-Management-Strategien und die Charakterisierungder erzielbaren Performance von sekundären Systemen.

Durch Spektrum-Sensing kann die Anwesenheit oder Abwesenheit der PUs de-tektiert werden. Die größten Herausforderungen hierbei sind die hohen Anforderun-gen an Empfindlichkeit, Zuverlässigkeit und Flexibilität, vor allem bei unvollständi-ger Kenntnis der Übertragungskanäle. Daher müssen die SUs die begrenzte a prioriKenntnis bzgl. der primären Übertragung effizient nutzen, um eine ausreichendeSensing-Performance zu erreichen. In der Dissertation wird hierfür das verallgemein-erte Likelihood-Ratio-Test-Framework auf kooperative Sensing-Probleme mit einerunbekannten Struktur des primären Signalraums und unbekannter Rauschvarianzenan den SUs angewandt. Die Effizienz der resultierenden Spektrum-Sensing Algorith-men für Fading-Kanäle wird auch in Bezug auf das „hidden primary user“-Problemgezeigt.

Basierend auf einer begrenzten Kenntnis der primären Übertragung, werden dieTransceiver-Strategien der SUs optimiert, um einen Kompromiss zwischen hohemsekundären Netzwerk-Durchsatz und der Beschränkung des Performance-Verlustsder primären Übertragung zu erreichen. Für Spektrum-Sharing-Systeme mit Einzel-Antennen-Geräten werden die Strategien zur Leistungsallokation an den SUs unterder Bedingung verschiedener „Quality of Service“ Einschränkungen für die primäreVerbindung untersucht. Es werden nicht nur optimale Strategien und nahezu opti-male Strategien geringer Komplexität entwickelt, sondern es wird auch die erzielbarePerformance des Systems näherungsweise in geschlossener Form angegeben. Zusät-zlich werden für Mehrantennen-Spektrum-Sharing-Systeme effiziente Transceiver-Str-ategien unter Berücksichtigung der unvollkommenen Kanalkenntnis entwickelt. Die

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Robustheit, die Optimalität und das Konvergenzverhalten der verschiedenen vorge-schlagenen Algorithmen werden quantitativ bestätigt und verglichen.

Die essentielle „kognitive“ Eigenschaft von DSA in Cognitive Radio Networksbesteht aus zwei Aspekten: Das Erfassen nützlicher Informationen aus der Umweltund die Nutzung dieser Informationen, um die spektrale Effizienz zu erhöhen. Dieswird beispielhaft mit einer Studie eines Hybrid-Paradigmas demonstriert, bei dem dieSUs die Sendeleistung in Abhängigkeit der Spektrum-Sensing-Informationen und vonOrts-Informationen anpassen. Im Vergleich zu den Standard-Paradigmen, d.h. deropportunistischen Übertragung oder der gemeinsamen Nutzung des Frequenzspek-trums ohne Sensing, erreichen die vorgeschlagenen Strategien im Hybrid-Paradigmaeine bessere Performance.

Contents

1 Introduction 1

1.1 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Spectrum Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Inteference Mitigation . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Cognitive Radio Systems 7

2.1 Network Paradigms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Interweave Paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.2 Underlay Paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.3 Overlay Paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Spectrum Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 Local Spectrum Sensing . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.2 Cooperative Spectrum Sensing . . . . . . . . . . . . . . . . . . . . 15

2.3 Dynamic Resource Allocation . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.1 Power Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.2 Transceiver Optimization . . . . . . . . . . . . . . . . . . . . . . . 19

3 Spectrum Sensing with Limited A Priori Information 23

3.1 GLRT Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2 Spectrum Sensing with Rank Information . . . . . . . . . . . . . . . . . . 24

3.2.1 Unknown Noise Variances . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.2 Known Noise Variances . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.3 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3 Spectrum Sensing Under Fading Channels . . . . . . . . . . . . . . . . . 32

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ii CONTENTS

3.3.1 Approximated Likelihood Ratio Test . . . . . . . . . . . . . . . . . 35

3.3.2 Energy Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.3.3 GLRT-Based Detection . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.3.4 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . 39

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4 Power Allocation with Partial Primary CSI 43

4.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2 Power Control with Interference Power Constraint . . . . . . . . . . . . . 45

4.2.1 Known CSI of the PT-SR Link . . . . . . . . . . . . . . . . . . . . . 45

4.2.2 Unknown CSI of the PT-SR Link . . . . . . . . . . . . . . . . . . . 51

4.3 Power Control with Outage Probability Constraint . . . . . . . . . . . . . 66

4.3.1 Optimal Power Allocation . . . . . . . . . . . . . . . . . . . . . . . 67

4.3.2 Suboptimal Power Allocation . . . . . . . . . . . . . . . . . . . . . 72

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5 Robust Transceiver Optimization 81

5.1 Bounded CSI Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.2 Stochastic CSI Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.2.1 SOCP-based Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 93

5.2.2 Downlink-Based Dual-Loop Algorithm . . . . . . . . . . . . . . . 94

5.2.3 Duality-Based Dual-Loop Algorithm . . . . . . . . . . . . . . . . . 99

5.2.4 Performance Comparison . . . . . . . . . . . . . . . . . . . . . . . 103

5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6 Sensing-Based Power Allocation 109

6.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.2 Soft-Decision Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.2.1 Optimal Power Allocation . . . . . . . . . . . . . . . . . . . . . . . 111

6.2.2 Suboptimal Power Allocation . . . . . . . . . . . . . . . . . . . . . 114

6.3 Hard-Decision Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.3.1 Optimal and Suboptimal Power Allocation . . . . . . . . . . . . . 115

6.3.2 Referenced Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . 117

CONTENTS iii

6.4 Interference Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

7 Conclusions and Outlook 127

A Mathematical Derivations 131

A.1 Solutions of [µi]k and [Σi]k,k′ . . . . . . . . . . . . . . . . . . . . . . . . . . 131

A.2 Analytical Form of Qavg(v) and Q′avg(v) . . . . . . . . . . . . . . . . . . 132

A.3 Proof of Proposition 4.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

A.4 Monotonicity of G(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

A.5 Monotonicity of H(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135



A.8 Strong Duality for the Optimization Problem (4.93) . . . . . . . . . . . . 140


A.10 Feasible Region Ft for x1,2(t, v) . . . . . . . . . . . . . . . . . . . . . . . . 141

A.11 Proof of Lemma 4.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

A.12 Proof of Lemma 4.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145



A.15 Solution of ∆Gk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

A.16 Solution of the Projection Operation . . . . . . . . . . . . . . . . . . . . . 149

A.17 Solution of gupp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

B Notation 151

Acronyms 153

Bibliography 155

iv CONTENTS

Chapter 1

Introduction

Wireless networks are currently assigned by spectrum regulatory bodies to licenseholders for exclusive usage. Due to the ever-increasing demand for wireless systemsand services, such static spectrum allocation policy yields a severely limed availablewireless spectrum. In contrast, recent observations and measurement studies of someregulatory agencies such as the Federal Communications Commission (FCC) haveshown that most of the radio frequency spectrum was inefficiently utilized [1]. Theparadox between spectrum scarcity and spectrum under-utilization motivates the ini-tial idea of cognitive radios (CRs) which enable dynamic spectrum access (DSA) inlicensed bands [90].

A CR is an intelligent device that is aware of the environment, adapts its transceiverparameters accordingly, and transmits in licensed bands with limited interference tolicensed users. Due to the lack of priority to access the licensed spectrum resources,unique challenges are imposed to enable DSA, such as interference avoidance with li-censed networks, quality of service (QoS) awareness for dynamic and heterogeneoustransmission, and seamless communication irrelevant to the variations of the licensedusers’ activity [5, 6]. To address these challenges, two main characteristics of CRs areintroduced in [55]

• Cognitive capability: A CR system is able to obtain the information from its radioenvironment and keep on tracking its variations. Such information is used tocharacterize favorable conditions in the environment and choose appropriateoperating parameters for the CR devices.

• Reconfigurability: A CR system is able to be dynamically programmed concern-ing its software or hardware settings based on the transmission decisions madeaccording to distinct scenarios.

The CR users are usually referred to as the secondary users (SUs), while the incumbentusers in the licensed spectrum band are termed the primary users (PUs).

New functionalities in a spectrum management framework are required for CRnetworks, such as spectrum sensing, spectrum decision, spectrum sharing, and spec-trum mobility [5]. These functions are realized through a cross-layer design approach.Since the focus of this research is on the PHY layer design of CR systems, two criticalfunctions are considered. First, spectrum sensing which indicates that a CR systemneeds to monitor the available spectrum in time and space. Based on the observa-tions, the spectrum opportunities are identified and their characteristics are captured.Second, spectrum sharing which means that a CR system is required to coordinate the

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2 Chapter 1. Introduction

transmission and the access to licensed bands, and to avoid severe interference toother users.1

Although the initial concept of CR systems is to enable the DSA in licensedbands [2], this idea can also be applied to the unlicensed bands, e.g., in the 2.4 GHzunlicensed band which possibly hosts systems like Bluetooth, Wi-Fi, etc [38]. The “in-telligence” of the SUs enables them to be aware of the spectrum occupancy and adapttheir transmission in accordance with informative knowledge learned from the envi-ronment. Hence, less interference and improved spectral efficiency are envisioned.

The standardization of CR systems has been recently carried on by several organi-zations, including the International Telecommunication Union (ITU), the Institute ofElectrical and Electronics Engineers (IEEE), the European Telecommunications Stan-dards Institute (ETSI), etc. For example, the IEEE 802.22 Wireless Regional Area Net-work (WRAN) standard allows the usage of CR techniques in the very high frequencyand ultra high frequency (VHF/UHF) bands licensed to television (TV) broadcastingservices and other coexisting services such as wireless microphones. The work [31]provides an overview of recently developed international standards related to CRsystems.

1.1 Research Objectives

DSA is a promising technology to greatly improve the spectrum efficiency in wirelesscommunication systems. However, new challenges encompass the design of radiodevices with “full cognition”. The essential requirement is to properly respond tothe interaction with the operating environment and avoid the harmful interference toboth licensed and unlicensed users. Hence, the ultimate objective of this research is toinvestigate several key interference management techniques in the DSA environment.More specifically, we focus on addressing the following topics.

1.1.1 Spectrum Sensing

Spectrum sensing is a basic functionality for the realization of the CR technologythrough interference avoidance. Driven by the initial motivation of CR systems toopportunistically use the available licensed bands, spectrum sensing identifies thespectrum access opportunities for the SUs to avoid interfering with the PUs’ trans-mission. Such opportunities can be characterized by the PUs’ activity information.The main challenge of spectrum sensing is a rigorous requirement on agility, sensi-tivity, and reliability, i.e, the SUs need to detect the PUs’ activity statuses quickly ina low signal to noise ratio (SNR) region but with high accuracy. For example, theIEEE 802.22 WRAN standard requires the detection of the incumbent TV broadcast-ing signals for power as low as −116 dBm and wireless microphone signals for power

1 Here, the meaning of spectrum sharing is slightly different from the definition in [5] since it is notrestricted to spectrum sharing only between the CR users but also allows for the coexistence withprimary systems.

1.1. Research Objectives 3

as low as −107 dBm. The sensing procedure should be finished in less than 2 secondswith at least 90% probability of detection and at most 10% false alarm rate [3, 118].Such requirements become even more challenging when facing the hidden primaryuser problem: for example, if a SU is located in a deep fade of the primary transmis-sion, it might cause unwanted interference to the PU due to a missed detection of thePUs’ transmission [139]. One effective method, termed cooperative spectrum sensingthat combines the sensing observations from multiple SUs, ameliorates the sensingperformance by exploiting more degrees of freedom (DoFs) of the SUs.

The environment uncertainties at the SUs, including the fading effect of the chan-nels and the noise uncertainty [113, 124], may severely degrade the sensing perfor-mance, even in a cooperative manner. Furthermore, a complete a priori knowledgefor detection, e.g., the channel state information (CSI) and the information concerningthe primary signals, is hard to be perfectly obtained at the SUs. In order to addressthese problems, the focus of the present investigation lies in the effective exploita-tion of the available but limited information at the SUs to optimize the cooperativespectrum sensing approaches.

1.1.2 Inteference Mitigation

Apart from opportunistic transmission, spectrum sharing, which allows for the con-current transmission of the PUs and the SUs, also has the potential to achieve highspectral efficiency as long as the interference caused to the primary system and amongthe SUs is properly controlled [38, 146]. Moreover, spectrum sharing might occur inopportunistic transmission under a missed detection of the primary transmission.

The network model for spectrum sharing can be considered as interference chan-nels consisting of the users with a hierarchical priority. Thus, different from tra-ditional multiuser networks, the new challenge involved in DSA is the interferencemitigation between the primary and secondary systems. Due to limited cooperationbetween the PUs and the SUs, the SUs only have restricted knowledge of the pri-mary system which makes interference management more difficult. To deal with thisproblem, the CR system can perform dynamic resource allocation, such as the properselection of the frequency bands, adjustment of the transmit power, or transceiveroptimization, to achieve QoS requirements and constrain the interference.

As a basis to optimize the dynamic resource allocation, we need to choose ap-propriate performance measures to evaluate the performance degradation to primarysystems. The commonly used measure is the so called “interference temperature” [33]representing the received interference power at the primary receivers (PRs). Alterna-tively, limiting the primary rate loss is the ultimate goal in regulating the secondarytransmission. Thus, the performance loss of the PUs can also be assessed with rate-related measures, e.g., an outage probability of the primary link. According to thetype of the available CSI at the SUs, the aforementioned metrics can be constrainedeither in a long-term or a short-term manner.

Power control is a simple but very effective way for interference mitigation. We areinterested in assessing the achievable performance regarding the secondary transmis-


sion given different QoS constraints on primary systems and limited a priori knowl-edge at the SUs. From a practical point of view, low-complexity and efficient near-optimal strategies are also desirable. In addition, in realistic systems, the PRs aresometimes passive receivers or mobile receivers, e.g., in TV broadcasting systems orin mobile cellular networks, respectively. Hence, their location information is difficultto be obtained at the SUs. Such location uncertainty is also required to be taken intoconsideration in the interference modeling. Furthermore, sensing results can also beincorporated in the design of the power allocation.

Nowadays, the use of multiple antennas at the transceivers is well established dueto the enhancement of the spectral efficiency by exploiting the spatial domains. Bysteering the directional transmission at the targeted users and avoiding it towards theinterfering users, transceiver optimization achieves the tradeoff between interferencemitigation and improving the performance of the desired links. In reality, perfectCSI of the entire network is hard to obtain at the SUs. Considering this effect in thealgorithmic design usually results in intractable non-convex optimization problems.Hence, one focus of this thesis is to develop efficient robust transceiver optimizationalgorithms in cognitive networks.

1.2 Outline

The target of this thesis is to investigate interference management techniques in CRnetworks with focuses on cooperative spectrum sensing strategies and interferencemitigation strategies for spectrum sharing. On the one hand, regarding cooperativesensing, we study the effective utilization of the limited a priori knowledge regard-ing the primary transmission and noise variances at the SUs. On the other hand,concerning interference mitigation in spectrum sharing systems, we consider exploit-ing dynamic resource allocation while taking into account the realistic challenge thatthe SUs only have partial CSI related to the primary transmission. Specifically, insingle-antenna spectrum sharing systems, the relation between the achievable perfor-mance of the secondary transmission and different QoS constraints of the primarytransmission is addressed. To further exploit the spatial DoFs introduced by usingmultiple-antenna transmission, we proceed to develop efficient transceiver optimiza-tion strategies for interference mitigation. Finally, the question on how to explore thelearning functionality in the design of transmission strategies at the SUs is answeredby an exemplified investigation on sensing-based power allocation algorithms.

The structure of the thesis is as follows. In Chapter 2, three paradigms of CRsystems are introduced: the interweave, the underlay, and the overlay paradigm. Brieflyspeaking, the interweave paradigm aims at guaranteeing the exclusive transmissionof the PUs and the SUs, the underlay paradigm mandates the simultaneous transmis-sion of both kinds of users, whereas the overlay paradigm also permits the concurrenttransmission of both kinds of users by allowing the SUs to use part of the secondarytransmission resources to relay the primary message. The preliminary assumptionsand the a priori information required for each paradigm are addressed, followed by a

1.2. Outline 5

brief discussion on the realistic solutions to deal with the respective challenges. Basedon the discussion, spectrum sensing and dynamic resource allocation are shown to betwo key techniques for interference management in CR systems. Therefore, we reca-pitulate commonly-used spectrum sensing strategies with a focus on the approachesbased on limited a priori knowledge. Additionally, ideas and abstract models for dy-namic resource allocation in CR systems are addressed including a detailed literaturereview.

Cooperative spectrum sensing algorithms with limited a priori information re-garding the primary transmission and noise variances are elaborated in Chapter 3.We are interested in how to exploit the available but limited information to improvethe sensing performance. Since such problems are modeled as binary hypothesis test-ing problems with unknown parameters, the generalized likelihood ratio test (GLRT)framework is applied to derive the solutions. Firstly, assuming an unknown structureof the primary signal space at the SUs, we exploit the rank information of the pri-mary signal space, which represents the DoFs of signals, to estimate its structure anddesign the GLRT-based methods. Secondly, we consider the influence of the fadingsensing and reporting channels on the cooperative sensing. The underlying assump-tion is that the SUs only have partial CSI of the sensing channels and do not knowthe structure of the primary signal space. The GLRT-based algorithms are proposedunder such circumstances to ameliorate the performance degradation caused by en-vironment uncertainties, e.g., the fading effects and noise variance uncertainty.

In Chapter 4, power allocation strategies for the secondary transmission are in-vestigated subject to different QoS constraints on the primary link. The goal is tocharacterize the connection between the achievable gain of the SUs and the tolerableperformance loss of the PUs. Assuming only partial CSI related to the PR is avail-able at the SU, we study the power allocation problems constrained by two kindsof constraints on the primary transmission: a conventional interference temperature(IT) constraint and an outage probability constraint. We not only address the optimalpower allocation strategies, but also design computationally efficient near-optimalstrategies. Moreover, the achievable performance of the system is approximately as-sessed.

Another dynamic resource allocation method, namely transceiver optimizationis presented in Chapter 5. We consider a cognitive downlink transmission withone multi-antenna secondary base station (BS) and multiple mobile SUs. Robusttransceiver filters are designed considering both of the following CSI error models:the bounded CSI error and the stochastic CSI error model. Due to the non-convexityof the problem, we resort to several distinct approaches for the development of ef-ficient algorithms, such as alternating methods, the constrained gradient projectionmethod, and most importantly, a method exploiting the uplink-downlink duality inthe network. Furthermore, the performance of different algorithms is quantitativelycompared.

Up to this point, spectrum sensing and interference mitigation strategies are inves-tigated for a single paradigm. In Chapter 6, we consider a hybrid paradigm combin-ing the interweave and the underlay paradigm. Specifically, the design of the power


allocation strategies in Chapter 4 is extended to additionally incorporate the reliabil-ity of the sensing outcome. Both soft-decision and hard-decision sensing results areexplored. Moreover, we consider the location uncertainty of the primary network andprovide a model of interference caused by the secondary transmission. The interfer-ence modeling result is also integrated into the power allocation problem. To thisend, we exemplify the performance gain from “cognition” in CR systems by assess-ing the achievable performance of the SUs through the effective utilization of sensingobservations.

Finally, Chapter 7 concludes the thesis and provides an outlook on potential futurework.

Chapter 2

Cognitive Radio Systems

The emergence of cognitive radio has the potential to increase the spectral efficiencyin traditional wireless communication systems where an exclusive spectrum usageis granted among different kind of users. Figure 2.1 exemplifies a network modelin which both the PUs and the SUs coexist. The fundamental requirement for thedesign of the SUs’ transceiver strategies is to limit the disturbance to primary systems.Based on different a priori information regarding the environment, cognitive radiosystems mainly employ three paradigms for the DSA: the interweave, the underlay,or the overlay paradigm. Different approaches are used in different paradigms forthe interference management, among which we focus on the two main perspectives.On the one hand, spectrum sensing, which characterizes the spectrum usage andthe PUs’ activity in the time, space, and frequency domains, can be considered as thebasic learning function of the SUs and it is widely utilized in the interweave paradigm.Based on such information, the SUs adapt their transmission strategy, e.g., suspendingthe transmission or lowering the transmit power to avoid the interference to the PUs.On the other hand, dynamic resource allocation, e.g., power allocation and transceiveroptimization, can effectively mitigate the interference when concurrent transmissionof the PUs and the SUs is allowed. It can be realized by exploiting the available sideinformation or the DoFs offered by the multi-user multiple-antenna systems.

In this chapter, we first briefly introduce three CR paradigms in Section 2.1. Twomain techniques of the interference management applicable to CR systems are revised:spectrum sensing in Section 2.2 and dynamic resource allocation in Section 2.3. Thesetwo approaches provide the basis of the remainder the thesis.

2.1 Network Paradigms

There are three main paradigms of the CR system: the interweave, the underlay, andthe overlay paradigm [38]. These paradigms are distinguished by the approaches tocooperate with the PUs. In the following, we recapitulate each of them, including thediscussion on the corresponding requirements and the key techniques for interferencemanagement.

2.1.1 Interweave Paradigm

The interweave paradigm, also named opportunistic spectrum access (OSA), representsthe opportunistic transmission of the SUs based on a priori knowledge of the availableradio resource assigned to the PUs. The idea is based on the original motivation of

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8 Chapter 2. Cognitive Radio Systems

Primary Network Secondary Network

Interference

Primary Base Station

Primary User

Secondary Base Station

Secondary User

Figure 2.1: Coexistence of a primary and a secondary infrastructure-based network.

CR that the spectrum resource is under-utilized as shown by various measurementsurveys, e.g., in [1,134]. Hence, spectral efficiency is improved by allowing the SUs toopportunistically access the resource when it is unoccupied by the PUs.

The preliminary for the SUs is to obtain the activity information of the PUs inspace, time, and frequency [125]. To enable this requirement, the SUs can make useof the geo-location databases, beacon signals, or spectrum sensing [139]. Amongthese strategies, spectrum sensing embodies mostly the “cognitive” concept that theCRs need to intelligently learn the available side information from the environment.Moreover, spectrum sensing requires minimal infrastructure changes of the PUs’ sys-tems. Therefore, we concentrate on spectrum sensing methods in this paradigm.Through periodical monitoring and detecting of the occupancy status of the spec-trum resources, the SUs transmit over the vacant resources, without yielding severeperformance degradation to the PUs.

Although the original understanding of the spectrum sensing is the characteri-zation of the spectrum opportunities, its general meaning also involves determiningother characteristics of the spectrum usage, such as user activity and channel statistics,modulation schemes of the signal, carrier frequency, and bandwidth, etc. Basically,spectrum sensing exemplifies the typical learning functionality of CR systems.

2.1.2 Underlay Paradigm

The underlay paradigm, also entitled spectrum sharing, permits the concurrent trans-mission of primary and cognitive users as long as the interference at the PUs is prop-erly limited. In this setting, the SUs need to have the knowledge of the potentialperformance degradation caused to the PUs, e.g., the interference power at the pri-mary receivers or the outage probability of the primary transmissions. In order tocontrol the interference, the initial idea is based on techniques of spread spectrum

2.2. Spectrum Sensing 9

and ultra-wideband (UWB) communication in which the signals from the SUs arespread below the noise level. However, such techniques may lead to conservativetransmit strategies of the SUs due to very low transmit powers. Alternatively, severalinterference mitigation techniques, such as adaptive power allocation or transceiveroptimization, allow for a control on the interference by exploiting the available CSIor multiple antennas at the SUs. Therefore, interference mitigation has raised a lot ofresearch interest in this paradigm.

2.1.3 Overlay Paradigm

Similar to the underlay paradigm, the overlay paradigm also allows for the concur-rent transmission of primary and cognitive users. The difference w.r.t. the underlayparadigm is that the interference to the primary transmission is compensated by usingpart of the secondary transmission to relay the primary message. To make it feasible,the underlying information required at the SUs is the knowledge of the PUs’ code-books, messages, and the channel gains. With such information, the SUs are able tocancel or mitigate the interference, thus, enhance spectral efficiency of the network.However, the practical realization is very challenging since it demands the SUs to havethe most a priori PUs’ knowledge among the three paradigms, especially the code-book and messages of one specific user type might be confidential for other kinds ofusers.

To summarize, the interweave paradigm aims at guaranteeing the exclusive trans-mission of the PUs and the SUs, whereas the underlay and the overlay paradigmsmandate the simultaneous transmission of both kinds of users. Due to the practicaldifficulty in realizing the overlay paradigm, we focus on the investigation of the firsttwo paradigms. Moreover, hybrid paradigms combining parts of the three paradigmsare also possible. For instance, the works [102, 107, 117] showed that an enhancedperformance is achieved by the SUs when jointly considering the interweave and theunderlay paradigm.

Since the ultimate requirement for the cognitive transmission is to ensure thatthe performance degradation to the PUs is properly controlled, we outline differ-ent interference management strategies used in different paradigms. Specifically, theinterweave paradigm requires the activity information of the PUs which can be ob-tained through spectrum sensing, whereas the underlay and overlay paradigms re-quire proper interference mitigation and cancelation approaches.

2.2 Spectrum Sensing

Spectrum sensing refers to the task of characterizing the spectrum usage and theactivity information of the PUs. Incorporating noise uncertainty, multi-path fading,and the shadowing effect into the specification, the SUs are required to quickly iden-tify the presence of the primary signal in the very low signal-to-noise ratio (SNR)region [113, 124]. In the IEEE 802.22 WRAN standard, the SUs should be able to


sense the incumbent TV broadcasting transmission for power as low as −116 dBm,and wireless microphone transmission for power at −107 dBm in less than 2 sec-onds [3, 118]. Such requirements regarding the sensitivity, agility, and reliability ofthe sensing are challenging for local spectrum sensing techniques. Furthermore, thehidden primary user problem might degrade the sensing performance by a singleSU [139]. For example, if a SU is obstructed from the primary transmit signal, itmight cause the unwanted interference to the PU due to a missed-detection of thePU’s transmission. To solve the hidden primary user problem, one effective approachis to exploit more DoFs in the network, e.g., making a joint decision based on multi-ple sensing observations collected from multiple SUs. Hence, cooperative spectrumsensing is desirable to improve the sensing performance. Comprehensive surveys onspectrum sensing have been conducted in [11, 139, 142].

In this section, we outline some commonly-used spectrum sensing strategies. Be-ginning with local sensing methods, including the conventional energy detection andsome feature detection methods, we proceed to the introduction of cooperative sens-ing which effectively ameliorates the local sensing subject to the hidden primary userproblem and thus increases the sensing reliability.

2.2.1 Local Spectrum Sensing

Local spectrum sensing is formulated as a binary hypothesis test problem as

y[n] =

{ω[n], H0

x[n] + ω[n], H1(2.1)

with ∀ n = 0, . . . N − 1. In eq. (2.1), x[n] represents the nth sample of the primarysignal and y[n] indicates its received version at the SU. The noise term at the nthsample is ω[n] which follows the proper complex Gaussian distribution. We assumethat ω[n] ∼ CN

(0, σ2I

)holds with the noise variance σ2. The dimensions of the

vectors x[n], y[n], and ω[n] are the same. We set the dimension equal to L, e.g., ifthere are L receive antennas at the SU. The observation interval includes N samples.The null hypothesis that the PU is absent is denoted by H0 and its presence is givenby H1. Stacking the vectorial observation of different samples into one vector, weobtain the compact form of the received data at the SU

y =[y[1]T, . . . , y[N]T

]T. (2.2)

The task of signal detection is to decide for the status H0 or H1. To achievethis goal, the test statistic Λ (y) is formed based on the observation data y and thencompared to a predefined threshold η [66]:

Λ (y)H1≷H0

η. (2.3)


The commonly used performance measures of the detector are the false alarm ratePFA that indicates the probability to declare H1 given the actual status H0 and theprobability of detection PD that represents the probability to declare H1 given thestatus H1:1

PFA = Pr {Λ (y) > η|H0} (2.4)PD = Pr {Λ (y) > η|H1} . (2.5)

In the IEEE 802.22 WRAN standard, the required PFA is 10 % and the required PD is90 % [118]. For the purpose of visualizing the performance, the receiver operatingcharacteristic (ROC) curve is used to show PD as a function of PFA. By choosingdifferent values of η, different operating points, i.e., different pairs of PD and PFAvalues, are selected along the ROC curves.

The essential part in the design of spectrum sensing algorithms is to obtain thetest statistic Λ (y) and the threshold η. According to the detection theory, they canbe designed, e.g., under the Neyman-Pearson (NP) or the Bayesian criterion [66, 103].Both result in the same form of the test statistic called the likelihood ratio

Λ (y) =p (y|H1)

p (y|H0)(2.6)

where p (y|Hi) denotes the probability density function (PDF) of the observation yunder the hypothesis Hi, i = 0, 1. In the following, we introduce several widely usedlocal spectrum sensing methods.

2.2.1.1 Energy Detection

One of the basic detection methods is known as energy detection (ED) [129]. Specif-ically, the received signal energy is measured over the observation interval. The ac-cumulated metric is compared to a predefined threshold to decide whether the PUs’transmission is active or not. The resulting test statistic is

ΛED (y) =‖y‖2

σ2

H1≷H0

η. (2.7)

The threshold η is set according to the system requirements, e.g., the fixed false alarmrate. If the primary signal is unknown to the detector and if signal samples are mod-eled as independent and identically distributed (i.i.d.) Gaussian random variables,the ED is the optimal detector according to the NP criterion.

The performance of the ED has been studied in [26] and sometimes can be ex-pressed in closed form. For example, if the transmit signal vector x[n] contains bi-nary phase-shift keying (BPSK) modulated signals with energy-per-symbol Eb, thetest statistic ΛED (y) follows a central chi-square distribution under the hypothesis1 For the simplification of illustration, herein we claim H0 for the special case that Λ (y) = η. In

practice, we can make equiprobable decisions ofH0 andH1 for Λ (y) = η. The performance measurescan be deduced analogously.


H0 with degree of freedom 2LN, while it follows a non-central chi-square distributionunder the hypothesis H1 with degree of freedom 2LN and non-centrality parameter2Eb/σ2. Closed-form expressions of PFA and PD can then be obtained using the cu-mulative density function (CDF) of the central or non-central chi-square distribution.Alternatively, if each element in x[n] is a complex Gaussian random variable with zeromean and variance σ2

s , e.g., this assumption approximately holds if the primary signalis an orthogonal frequency-division multiplexing (OFDM) signal [79], the test statis-tic ΛED (y) under H1 becomes a scaled central chi-square distribution with degree offreedom 2LN and scaling factor

(σ2

s + σ2)/σ2.The ED requires perfect knowledge of the noise variances. Due to the noise un-

certainty in practical systems, the ED cannot make an accurate decision given thatthe SNR is below a certain threshold even for infinitely long sensing durations. Suchan SNR threshold is called the SNR wall [124]. In fact, the uncertainty in the noisemodeling causes a performance degradation in any moment-based detector. To allevi-ate this drawback, the work [124] proposed noise calibration schemes to enhance thenoise modeling. Alternatively, the embedded features in the primary signals can beexploited to estimate the missing knowledge of the noise variances, rendering sensingalgorithms robust to the noise uncertainty [148].

2.2.1.2 Matched Filter

In practical systems, pilots or preambles are usually transmitted periodically with theprimary signals to help the receivers perform the channel estimation, synchronization,etc. If such information of the PUs is obtained at the SUs, e.g., provided by theprimary systems’ standard, it can be exploited to perform the matched filter (MF)-based sensing. In particular, if the primary signal vector s ∈ CLN×1 is fixed andknown to the SUs, the optimal approach according to the NP criterion for signaldetection is MF with the test statistic

ΛMF (y) = R(

yHs)

. (2.8)

The distribution of the test statistic for an additive white Gaussian noise (AWGN)channel can be analogously derived according to [66]. A example of its application inthe IEEE 802.22 WRAN standard was shown in [23].

The advantage of MF-based detection mainly consists of three aspects. First, it isable to distinguish the primary signals from interferences and noises. Second, it workswell even in the low SNR region. Third, compared to the ED, it requires a shorter ob-servation time to achieve the predefined performance target. However, this coherentdetection has its limitations. For instance, it requires the knowledge of the primarysignal at both the physical layer (e.g., pulse shaping and modulation type of the pilot)and the medium access control layer (e.g., synchronization). Furthermore, the SUs arerequired to perform timing, carrier synchronization, and even equalization to detecta dedicated kind of primary signals, which increases the implementation complexityof the sensing method.


2.2.1.3 Cyclostationary-based Detection

The principle of cyclostationary detection is to exploit the embedded periodicity of thestatistics of the man-made signals which may be caused by modulation and coding,the cyclic prefix, or intentionally inserted signals to aid the synchronization, channelestimation, or equalization. This kind of signals can usually be modeled as a cyclo-stationary random process. Defining the autocorrelation function of a discrete-timerandom process y[n] as

ry [n, τ] = E {y[n]y∗[n + τ]} (2.9)

the random process y[n] is wide-sense cyclostationary if ry[n, τ] is a periodic functionin n for any τ [32]. Mathematically, this feature of the observed signal can be extractedby analyzing the cyclic autocorrelation in the time domain or its equivalent Fouriertransformation called cyclic spectrum in the frequency domain

cyclic autocorrelation : Rαy(τ) =

1N

N−1∑n=0

ry [n, τ] e−j2παn (2.10)

cyclic spectrum : Sαy( f ) =

∑τ

Rαy(τ)e

−j2π f τ (2.11)

where α denotes the cyclic frequency and N is the observation length. Basically, thetest statistic of cyclostationary-based detection is a function of Rα

y(τ) and Sαy( f ). For

example, the cyclostationary-based feature detector using one cyclic frequency hasbeen proposed in [25] and lately extended to use multiple cyclic frequencies in [86].

In general, the cyclic features of different types of signals are distinct. Thus, themain advantage of cyclostationary-based detection is its capability of differentiatingbetween specific types of primary signals, interferences, and noises. Nevertheless,cyclostationary-based detection is vulnerable to model uncertainties and the channelfading effect [123]. Moreover, it is susceptible to sampling clock offsets [128].

2.2.1.4 Subspace-Based Detection

The idea of subspace-based detection is that the primary signal received at the SU isusually correlated, e.g., due to multiple receiver antennas or over-sampling effect. Onthe contrary, the noise at the SU is in general spatially and temporally white. Thisfeature can be exploited in spectrum sensing to distinguish the cases containing theexistence of the primary signals or pure noise, respectively. Due to the fact that sucha feature is extracted from the specific structure of the sample covariance matrix, thecorresponding algorithms are usually termed subspace-based detection methods.

In order to measure the signal whiteness, we choose some covariance based sta-tistical parameters to characterize the subspace features. Specifically, considering thesignal model in (2.1), the sample covariance matrix Ry of the received signal is

Ry =1N

N−1∑n=0

y[n]yH[n]. (2.12)


Let λi, ∀ i = 0, . . . L − 1 be the eigenvalues of Ry sorted in descending order of themagnitude. Several recent works designed the test statistics using these eigenvalues.For instance, maximum-minimum eigenvalue (MME) detection in [140] proposed thetest statistic as the ratio of the maximum and minimum eigenvalues

ΛMME(y) =maxi λi

mini λi

H1≷H0

η. (2.13)

The distribution of ΛMME(y) has been approximated using random matrix theory.Therefore, a decision threshold can be derived given that the false-alarm probabilityis fixed. Alternatively, the covariance based test statistic, e.g., the ratio of its diagonaland off-diagonal elements in Ry was designed in [141]. The asymptotic distributionof the ratio was also derived for a selected decision threshold.

In a practical scenario, a priori knowledge of the primary signal space and thenoise variance matrix might not be available at the SUs. In detection theory, the GLRTprinciple is a widely used approach to tackle the binary hypothesis testing problemwith unknown parameters [66]. Asymptotically, the GLRT test statistic is equivalent tothe uniformly most powerful (UMP) test among all invariant statistical tests [76]. Thedetailed introduction of the GLRT principle will be presented in Section 3.1. Recently,the GLRT framework has been applied to the context of developing the subspace-based spectrum sensing problem when only certain a priori information is available.For instance, The authors in [148] studied the case assuming spatially white noise withthe same variance at each receiver and the unknown signal space of arbitrary rank,while the authors in [122, 132] considered the similar problem but assuming that therank of the unknown signal space is equal to one. Considering the noise variances aredifferent at each receiver and the rank of the unknown signal space is one, the sensingstrategy was designed in [84]. Exploiting the information that the unknown OFDMsignal covariance matrix has known eigenvalue multiplicities, the GLRT framework isapplied in [9].

However, the potential utilization of the rank information has not been fully ad-dressed. The rank information which indicates the number of independent signalsources can be estimated by the methods in [75,133] or provided by the PU standard.For example, if sensing channels are mutually independent, the rank information isthe number of independent signal streams transmitted by the PU. The rank informa-tion is useful to extract the signal space embedded in the sample covariance matrixof the received signals, especially for the case of a rank-deficient signal space. Forexample, a sensing method with rank information of the primary signal under theassumption of equal noise variances at the SUs was studied in [16]. However, thesensing method in a more general scenario where the noise variances are distinct atmultiple SUs is not clear. This setting leads to the difficulty in deriving a GLRT-basedmethod since the closed-form maximum likelihood estimator (MLE) of the unknownparameters is mathematically intractable to the best of our knowledge.

In summary, subspace-based approaches exploit the correlated nature of the pri-mary signals which is in contrast to the whiteness of the noises. This feature is gen-


BS: base station

SU: secondary user

FC: fusion center

Primary BS

FC

SU1

SU2

SU3

Figure 2.2: Hidden primary user problem in spectrum sensing.

erally embedded inside the primary signals, or can also be artificially constructed,e.g., using multiple receive antennas at the SUs. Under the circumstance of limitedknowledge regarding the primary signal space and the noise variances, the GLRTprinciple can be applied to the subspace-based sensing methods to effectively tacklethe problem.

2.2.2 Cooperative Spectrum Sensing

The local sensing by a single SU is subject to multi-path fading and shadowing. Fig.2.2 depicts one example called the hidden primary user problem: the link from theprimary BS to the first SU is in a deep fading, i.e., obstructed by a large obstacle fromthe primary transmitter (PT). Consequently, it might cause unexpected interference tothe PU since the received primary signal power is too weak to detect. This uncertaintycan be effectively mitigated by the spatial diversity in cooperative sensing. Addition-ally, cooperative sensing can relax the sensitivity requirement of local sensing andincrease the agility of making a sensing decision.

The cooperative schemes are classified according to the cooperation manner: dis-tributed manner and centralized manner. On the one hand, in centralized sensing,there is a central unit existing in the network to collect the sensing observations fromthe SUs and make a common sensing decision. This sensing manner can be widelyapplied to the infrastructure-based secondary network where a cognitive BS or accesspoint (AP) serves as the fusion center (FC) to collect the sensing decisions from coop-erating SUs. On the other hand, in distributed sensing, the SUs share the informationamong each other and aim at reaching a final sensing decision. It is usually appliedin distributed networks, e.g., mobile ad hoc networks, where the users might ex-plore consensus-based methods [78]. Compared to the centralized manner, althougha distributed spectrum sensing system does not require the existence of the FC, an


Figure 2.3: Fusion-based cooperative sensing.

increased implementation complexity is required at each SU and overhead channelsare still needed to coordinate all the SUs. Therefore, we focus on the study of thecentralized cooperative sensing.

The structure of cooperative sensing is illustrated in Figure 2.3 assuming thereare K SUs in the network. Under the circumstance H0 or H1, the primary signal sis transmitted through the independent fading channels h1, . . . , hK to each user. Theadditive Gaussian noises of the sensing and reporting channels are represented bynk and vk, ∀ k = 1, . . . , K, respectively. Based on the local decision rule γk(·), ∀ k =1, . . . , K, the individual user k forwards its sensing decision uk = γk(xk) to the FCthrough the reporting channel gk and the noise vk. The FC accumulates the noisydecisions yk, ∀ k = 1, . . . , K, from all the sensors and finally makes the cooperativesensing decision based on the decision rule γ0(·).

Depending on the type of information collected at the secondary FC, cooperativesensing can be categorized into soft-decision fusion and hard-decision fusion.

• Soft-decision fusion: For the soft-decision fusion, each SU sends soft values, e.g.,received sensing data or energy, to the FC without quantization. Assuming thereceived signals from K cooperative SUs are mutually independent given H0 orH1, the log likelihood ratio (LRT) at the FC is

Λ0LRT = ln

(p (y1, . . . , yK|H1)

p (y1, . . . , yK|H0)

)= ln

( K∏k=1

p (yk|H1)

p (yk|H0)

)

=K∑

k=1

ln(

p (yk|H1)

p (yk|H0)

)

2.3. Dynamic Resource Allocation 17

=K∑

k=1

ΛkLRT (2.14)

where Λ0LLR is the LRT test statistic at the FC and Λk

LLR, ∀ k = 1, . . . , K, is theLRT test statistic at the kth SU. Eq. (2.14) indicates that the optimal fusion rule isthe sum of the local LRTs from all cooperative SUs under the mutual indepen-dence assumption. Correlation among the secondary signals can degrade thecooperative sensing performance [36].

• Hard-decision fusion: In order to reduce the signaling overhead between the SUsand the FCs, each SU can transmit the individual sensing decision by binary orBPSK signaling to the FC. The FC combines the hard decisions into one commondecision, e.g., by performing the classical AND, OR, or voting rules [77, 150].This scheme is named hard-decision sensing.

In general, soft-decision fusion outperforms hard-decision fusion at the expenseof a larger signaling overhead to forward the data. Alternatively, quantized decisionfusion is desirable to achieve the tradeoff between these two issues.

2.3 Dynamic Resource Allocation

Due to the hierarchical usage of the resources by primary and secondary systems,the main objective in the design of secondary systems is to constrain the performancedegradation caused to the PUs below some tolerable limit. Dynamic resource alloca-tion is widely considered as an effective way to compromise interference mitigationand improving the performance of secondary systems, cf. [147]. To achieve this goal,the SUs adjust the transceiver strategies, such as transmit power, bandwidth, pre-coders, and equalizers, according to the channel variations and the available a prioriinformation regarding the wireless communication environment. For instance, if theCSI of the secondary links is in a favorable status and the primary user is inactive, theSUs can “greedily” use the available resource for transmission, and vice versa. Mostimportantly, the optimization of the secondary transceiver parameters are constrainedby the performance degradation caused to the PUs. A commonly used performancemeasure is the received interference power at the PR termed the IT [33], either in thelong-term or short-term manner [146].

In this section, we concentrate on introducing two key techniques for dynamicresource allocation: power allocation and transceiver optimization. Their effectivenessin interference management has been extensively studied in conventional multiusercellular networks. Herein, we focus on their application in the DSA setting.

2.3.1 Power Allocation

In spectrum sharing networks, the secondary transmitters (STs) need to properlyadapt the power for the sake of enhancing the throughput of the secondary network


Figure 2.4: System model considered for power allocation.

and limiting the interference to the PUs. One example of the system model is de-picted in Figure 2.4. A single secondary link coexists with a single primary link wheneach transmitter or receiver is equipped with a single antenna. The methodology inthe power control design can also be applied to more sophisticated scenarios, e.g.,with multiple PUs and SUs. In order to protect the primary transmission, variousworks consider an IT constraint in the design of secondary transmit strategies. Forexample, Gastpar studied the achievable rate of the secondary network over addi-tive white Gaussian noise channels, subject to an average IT constraint [33]. In [37],Ghasemi and Sousa proposed the optimal power allocation strategies to maximizethe secondary achievable rate over different types of fading channels, considering ei-ther a peak or an average IT constraint. The work [64] studied the optimal powerallocation when jointly imposing a transmit power and an IT constraint. Consideringboth power constraints and incorporating the spectrum sensing result, the work [8]designed adaptive power and rate allocation schemes, and the work [120] studiedthe outage capacity under truncated channel inversion with a fixed rate strategy. Al-ternatively, limiting the primary capacity loss is the ultimate goal in regulating thesecondary transmission. To this end, Zhang investigated the optimal power controlunder the PU’s ergodic capacity-loss constraint in [149] and Kang et al. designed op-timal power allocation methods constrained by an outage probability of the primarylink in [65]. Both [149] and [65] demonstrated that a gain on the secondary rate isachieved by further exploiting the primary CSI instead of considering solely the ITconstraint. Inspired by such an observation, the authors in [81] studied the powerallocation problem to maximize the achievable rate of the secondary link subject tooutage probability constraints on both primary and secondary links, and an averagetransmit power constraint. However, the prerequisite of all the above works is that theST has instantaneous CSI of the entire network. In practice, this is challenging due tothe difficulty in the cooperation between the PUs and the SUs.

In reality, the SUs can only acquire partial CSI of the links to the PUs due to lim-ited cooperation with the PUs. Two kinds of partial CSI were considered for power


allocation design in spectrum sharing systems: outdated CSI [73, 89, 94, 110, 121] andstatistical CSI [12, 28, 80, 116]. On the one hand, assuming that the ST has only out-dated CSI from the ST to the PR link, the achievable rate of the secondary link wasanalyzed under an average interference power constraint in [94] or both a peak andan average power constraint in [110]. Nevertheless, the interference from the PT to thesecondary receiver (SR) was not considered in [94] and [110]. Later, power allocationin the cross-interfering spectrum sharing system was investigated under an averagepower constraint [121] with outdated CSI. The impact of outdated CSI on the achiev-able rate of the secondary link was also thoroughly investigated in [73,89], where thecorrelation of outdated CSI and the actual CSI was exploited to enhance the achievableperformance of the secondary link. It was found that the low correlation between theoutdated CSI and the actual CSI yields a large capacity loss of the secondary trans-mission. On the other hand, statistical CSI is desirable since it changes on a timescale that is much larger than the channel coherence time. Such CSI can be obtainedby exploiting some side information, e.g., location information [20, 61, 137]. Givensuch statistical CSI the performance loss of the PUs can be limited in an average ora probability-constrained manner. For example, the achievable rate of the secondarylink was investigated under an interference outage constraint in [80] and a minimumsignal-to-interference-plus-noise ratio (SINR) constraint of the PR [28]. However, theinterference from the PT to the SR was not considered in the above work. Taking thisinterference into account and assuming that the ST has perfect CSI of this interfer-ing link, the authors in [116] designed the optimal and suboptimal power allocationapproaches. Different CSI assumptions were made in [12] which studied power con-trol methods aiming at achieving the minimum required mean rate of the secondarylink subject to either a peak, an average transmit power constraint, or an outage con-straint. The authors assumed that the ST has instantaneous CSI of the ST-SR link andthe ST-PR link, but only statistical CSI of the PT-PR and the PT-SR link. Nevertheless,it is still challenging for the ST to obtain instantaneous CSI of the ST-PR link, since itrequires the PR to estimate the CSI and feed it back to the ST.

Furthermore, the relation between the achievable performance of the SUs and theperformance degradation of the PUs is desirable given certain power control strate-gies. The corresponding performance analysis can provide the insights on the benefitof exploiting the DSA and help to design guidelines in selecting system parametersfor power adaptation strategies. However, closed-form expression of performance isalways mathematically intractable. The alternative way is to develop a near-optimalstrategy and characterize its analytical performance. Thus, the result can be used as agood approximation of the optimal achievable performance.

2.3.2 Transceiver Optimization

Spatial DoFs are widely exploited by employing multiple-antenna systems to keepthe interference below a tolerable limit and maintain a high spectrum utilization.For instance, in traditional downlink cognitive systems, joint transceiver optimizationis performed by using precoding at the BS and linear equalization at each mobile


Figure 2.5: System model for transceiver optimization. Solid lines: desired links;dashed lines: interfering links.

terminal [114, 131, 135]. Specifically, a directional emission pattern is formed at thetransmitter towards the desired user with the optimized precoders, while keepingthe interference leaked to the undesired users low. Similarly, at the receiver side,interference and the effect of channel distortion are mitigated with the optimizedequalizers. Recently, transceiver optimization techniques have been studied in CRsystems, e.g., [60,143]. Figure 2.5 exemplifies the scenario assuming a multiple accesssecondary network with K STs and one SR coexisting with M PRs. Contrary to omni-directional transmission, the first ST steers the transmit vector to the SR and avoidsthe transmission to the PRs. Consequently, the throughput of the secondary networkis increased and the interference to the PRs is minimized.

Implementation of transceiver optimization requires CSI of the entire network atboth the transmit and receiver sides. In reality, due to imperfect channel estimation,outdated estimates for fast time-varying channels or the quantization effect in the lim-ited feedback, erroneous CSI is inevitable. Particularly in CR networks, the accuratechannel knowledge is hard to acquire due to the limited cooperation with PUs. Sincethe transceiver optimization is quite sensitive to imperfect CSI [96], robust designbecomes indispensable to deal with the channel uncertainties.

Two types of CSI error models are commonly considered: the bounded and thestochastic model. In general, the bounded model is applicable to frequency divisionduplex (FDD) systems, when the CSI error arises from the quantization of feedbackinformation and is related to the quantization regions; the stochastic model appliesto time division duplex (TDD) systems where the CSI is reused due to the reciprocity


of the uplink and downlink and its error is due to inaccurate channel estimation atthe transmitter. Two principles are generally used for the system design with CSIerrors. One is the worst-case approach to guarantee a certain system performance forany channel realizations within the CSI uncertainty region. For example, the authorsin [144] studied the system composed of one cognitive user and one PU, where CSIof the ST-SR link is perfectly known but erroneous CSI of the ST-PR link is boundedwithin an ellipsoid uncertainty region. In [35], Gharavol et al. investigated the robustdownlink beamforming design in multiuser multiple-input and single-output (MISO)cognitive networks, aiming at minimizing the transmit power of the secondary BSs,while targeting a lower bound on the received SINR of the SUs and imposing an upperlimit on the interference power constraint at the PRs. Meanwhile the work in [153]considered the same scenario but aimed at maximizing the minimum of the receivedSINR of the SUs. The CSI uncertainty model was generalized from the bounded ballregion to a bounded ellipsoid region. The other is the stochastic approach whichguarantees a certain average system performance over all channel realizations withinthe uncertainty region [151].

In summary, transceiver optimization is an effective way for interference mitiga-tion in CR networks. Due to practical difficulties, a robust design considering imper-fect CSI is imperative.

Chapter 3

Spectrum Sensing with Limited APriori Information

Spectrum sensing aims at detecting the presence or absence of the primary trans-mission. The main challenge is to detect the PUs’ activity in the low SNR region inwhich the environment uncertainties, e.g., fading channels and noise uncertainty, canseverely degrade the performance. Moreover, due to the limited cooperation betweenthe PUs and the SUs, the information regarding the primary signals and the CSI ofthe entire network is usually not available at the SUs. Driven by the aforementionedissues, the design of efficient sensing strategies which are robust to the uncertaintiesand can fully exploit the available but limited a priori knowledge is necessary.

Recently, subspace-based spectrum sensing methods have raised a lot of researchinterest due to its effectiveness over the conventional ED approach and robustness tothe environment uncertainties [16, 142]. However, these methods require the SUs toobtain complete a priori knowledge of the primary signal space and noise variances,which is unrealistic in practice. In detection theory, the GLRT principle is a widelyused approach to deal with binary hypothesis testing problems with unknown pa-rameters [66]. Since spectrum sensing is usually formulated as a binary hypothesistesting problem, the GLRT framework has recently been applied to solve the sensingproblem when only certain a priori information is available, e.g., in [9,84,122,132,148].

In this chapter, we investigate cooperative spectrum sensing algorithms with lim-ited a priori information. In order to provide a theoretical basis for later use, a briefreview of the GLRT principle is first provided in Section 3.1. Then, two detailed sens-ing problems are studied: First, in Section 3.2, we exploit the rank information toefficiently extract the primary signal space and design the corresponding subspace-based sensing algorithm. Herein, the signals through reporting channels are assumedto be perfectly recovered. Second, in Section 3.3, we consider that the signals throughsensing and reporting channels are subject to fading effects. Moreover, the structureof the primary signal spaces are unknown at the SUs. The GLRT-based sensing algo-rithms are derived by exploiting the available partial CSI of the channels.

The results presented in this chapter have been published in part by the authorin [39, 45, 46]1.

1 In reference to IEEE copyright material which is used with permission in this thesis, the IEEE doesnot endorse any of RWTH Aachen University’s products or services. Internal or personal use ofthis material is permitted. If interested in reprinting/republishing IEEE copyrighted material foradvertising or promotional purposes or for creating new collective works for resale or redistribution,please go to http://www.ieee.org/publications_standards/publications/rights/rights_link.html to learn how to obtain a License from RightsLink.

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24 Chapter 3. Spectrum Sensing with Limited A Priori Information

3.1 GLRT Principle

We consider a binary hypothesis testing problem with unknown parameter θ. Theobservation data x is used to make a choice between two competing hypotheses: thenull hypothesis H0 and the alternative hypothesis H1. We denote θ0 and θ1 as thevalues of the unknown parameter θ under H0 and H1, respectively. The MLEs of θ0and θ1 given as θ0 and θ1 are

θ0 = arg maxθ0

p(x|H0, θ0) (3.1)

θ1 = arg maxθ1

p(x|H1, θ1). (3.2)

GLRT-based methods rely on the conventional LRT method by replacing the exactknowledge of θ with the MLEs θ0 and θ1. Thus, the resulting test statistic is

ΛGLRT(x) = lnp(x|H1, θ1 = θ1)

p(x|H0, θ0 = θ0). (3.3)

Consequently, the solution of the binary hypothesis problem is determined as

ΛGLRT(x)H1≷H0

γGLRT (3.4)

where γGLRT is the pre-defined threshold. The decision in (3.4) indicates that wesimply determine the outcome with the largest probability given the observation x.

3.2 Spectrum Sensing with Rank Information

In a practical cognitive radio system, the exact knowledge of the primary signal spaceand noise variances is hard to obtain. Although several works investigated this kindof spectrum sensing problems [9,84,122,132,148], the potential utilization of the rankinformation of the primary signal space has not been fully addressed. The rank in-formation indicates the number of independent signal sources. For example, if thechannels of all sensing links are mutually independent, the rank information is thenumber of independent signal streams transmitted by the PUs. Such informationcan be estimated by the methods in [75, 133] or provided by the standard of primarysystems. In what follows, we attempt to show how to use the rank information toextract and identify the signal space embedded in the sample covariance matrix ofthe received signals. With the estimated signal space at hand, GLRT-based sensingmethods are then designed.

We consider a cooperative CR network with M SUs. Each SU is equipped with asingle receive antenna and performs spectrum sensing based on the observation overN sampling times. Denoting the stacked receive signals of M SUs at the nth time

3.2. Spectrum Sensing with Rank Information 25

instant as x[n] ∈ CM×1, ∀ n = 1, . . . , N, the task is to differentiate binary hypotheseswhere the absence and the presence of the PU are given by H0 and H1, respectively

H0 : x[n] = w[n]H1 : x[n] = Hq[n] + w[n]

where the primary signal at the nth time instant is denoted by q[n] ∈ Cq×1. The CSI ofthe flat fading sensing channels is H ∈ CM×q. The received signal Hq[n] is temporallywhite and Hq[n] ∼ CN (0, RS) follows, where RS indicates its covariance matrix. Thespatially and temporally white noises received at the SUs are given by w[n] ∈ CM×1

and w[n] ∼ CN (0, Λ) withΛ = diag{σ2

1 , . . . , σ2M} (3.5)

where the noise variance at the nth SU is given by σ2m, ∀m = 1, . . . , M. We consider

a general case that Λ has different diagonal entries due to the random layout ofcooperative SUs in the network.

Defining x =[xT[1], . . . , xT[N]

]T, its PDF under Hi is given as

p(x|Hi) =1

πMN (det Σi)N exp

(−Ntr

{Σ−1

i Rx

})(3.6)

where Σi is represented the covariance matrices of x under both hypotheses. Thus,we have

Σ0 = Λ (3.7)Σ1 = RS + Λ (3.8)

and the sample covariance matrix of x is

Rx =1N

N∑n=1

x[n]xH[n]. (3.9)

In practice, the noise variance can be calibrated at each SU. However, due tothe noise uncertainty effect [124], the calibration error would deteriorate the sensingperformance. In the following, we apply the GLRT principle to both scenarios as-suming that the noise covariance matrix is known and unknown. In both scenarios,the structure of primary signal space is unknown at the SUs. Nevertheless, its rankinformation

rank(RS) = r, 0 ≤ r ≤ M (3.10)

is known at the SUs.


3.2.1 Unknown Noise Variances

In this case, both covariance matrices Σi, i = 0, 1 of the received signal x under H0and H1 are unknown. Therefore, the main work in applying the GLRT principle is tocalculate the MLEs of the unknown parameters.

Using the PDF expression in (3.6), the log-likelihood function (LLF) of x under thehypothesis H0 is

ln p(x|H0, Λ) = −MN ln π − N ln det Λ− Ntr{Λ−1Rx}. (3.11)

where we use (3.7) since here the received signal x is only composed of the noise term.Applying (3.1), the MLE of Λ is obtained as

Λ|H0 = diag{Rx}. (3.12)

Similarly, under H1, the LLF of x is

ln p(x|H1, RS, Λ) =−MN ln π − N ln det(RS + Λ)− Ntr{(RS + Λ)−1 Rx

}. (3.13)

Using

RS = Λ−1/2RSΛ−1/2 (3.14)

Rx = Λ−1/2RxΛ−1/2 (3.15)

Eq. (3.13) is reformulated to

ln p(x|H1, RS, Λ) =−MN ln π − N ln det(IM + RS)

− N ln det Λ− Ntr{(IM + RS)

−1 Rx

}. (3.16)

Note that the rank of RS is equal to that of RS.To apply the GLRT principle, we need to calculate the MLE of RS. This can be

obtained by maximizing the LLF in (3.16). The result is given in the following lemma.

Lemma 3.2.1. Consider the optimization problem

maxRS

− ln det (RS + IM)− tr{(RS + IM)

−1 Rx

}(3.17)

s.t. RS � 0

where the rank of RS is r. The optimal solution of (3.17) is denoted by RS and given as

RS =

m1∑i=1

(λx,i − 1

)+ ux,iuHx,i (3.18)


where λx,i, ∀ i = 1, . . . , M are the eigenvalues of Rx in descending order of the magnitudeand ux,i are the corresponding eigenvectors. Assuming r1 is the largest index of λx,i whichsatisfies λx,i ≥ 1, then we have m1 = min(r, r1).

Proof. We first rewrite the two terms in the objective function of (3.17) which containRS as

det(RS + IM) =r∏

i=1

(λs,i + 1) (3.19)

(RS + IM)−1 = US(ΛS + IM)−1UHS (3.20)

where US = [us,1, . . . , us,M] is a unitary matrix containing eigenvectors of RS. Thediagonal matrix ΛS contains the eigenvalues of RS in descending order:

ΛS = diag{

λs,1, λs,2, . . . , λs,r, 0, . . . , 0}

. (3.21)

Considering the positive semidefiniteness (PSD) of matrix ΛS and its rank is equal tor, it implicitely requires that

λs,i > 0, ∀ i = 1, . . . , r. (3.22)

Taking (3.19) and (3.20) into the objective function of (3.17) yields

RS = arg minUS,ΛS�0

f (US, ΛS) (3.23)

with

f (US, ΛS) =r∑

i=1

ln(λs,i + 1) + tr{

US(ΛS + IM)−1UHS Rx

}. (3.24)

To solve the optimization problem (3.23), we first consider the optimization of ma-trix US which contain M orthogonal normalized column vectors. The optimal US tominimize the function f (US, ΛS) can be derived according to [7, Theorem 2] as

us,i = ux,i, ∀ i = 1, . . . , M. (3.25)

Next, integrating (3.25) into f (US, ΛS) in (3.24), (3.24) is simplified as a function ofΛS only. The resulting new objective function g (ΛS) is

g (ΛS) =r∑

i=1

(ln(λs,i + 1

)+

λx,i

λs,i + 1

). (3.26)

Considering a new optimization variable

λs,i =(λs,i + 1

)−1 , ∀ i = 1, . . . , r (3.27)


we have the following property of λs,i due to the positivity of λs,i in (3.22)

λs,i < 1, ∀ i = 1, . . . , r. (3.28)

Using (3.27), Eq. (3.26) is written as

g(ΛS)=

r∑i=1

(− ln λs,i + λx,iλs,i

)(3.29)

whereΛS = diag

{λs,1, λs,2, . . . , λs,r, 0, . . . , 0

}. (3.30)

It is easy to show that g(ΛS)

in (3.29) is a convex function of λs,i, ∀ i = 1, . . . , r. Takingthe derivative of the above equation over each λs,i and setting it to zero, we obtain theestimate of λs,i as λ−1

x,i . Recalling the condition in (3.28), we have

ˆλs,i = min(

λ−1x,i , 1

), ∀ i = 1, . . . , r. (3.31)

Consequently, the MLEs of λs,i is obtained from (3.27) and (3.31) as

ˆλs,i = (λx,i − 1)+, ∀ i = 1, . . . , r. (3.32)

Combining (3.25) and (3.32) yields (3.18). This concludes the proof.

Remark: The similar problem considered in Lemma 3.2.1 for Λ = I was discussedin [16]. However, our result is different from that of [16] where m1 is equal to r. Weadditionally consider the PSD characteristic of RS and choose m1 = min (r, r1).

The LLF in (3.16) is equal to the objective function in (3.17) plus a term that isindependent of RS. Therefore, the MLE of RS is given in (3.18). Integrating this resultinto (3.16), we get the LLF as a function of only one unknown parameter Λ since Rxis a function of Λ

ln p(x|H1, Λ) = −MN ln π − N ln det Rx

− NM∑

i=m1+1

λx,i + N lnM∏

i=m1+1

λx,i − Nm1. (3.33)

The optimal λx,i to maximize (3.33) is

λx,i = 1, ∀ i = m1 + 1, . . . , M.

However, the optimal Λ to maximize (3.33) is hard to obtain [84] under this circum-stance. Follow the idea in [84], we use the near-optimal MLE in the low SNR regionof the received primary signals

Λ|H1 = diag{Rx}. (3.34)


From Lemma 3.2.1, we see that ˆRS is a function of Λ. However, the perfect informationof Λ is not available in this case. Thus, we represent the MLE of RS by replacing Λ inˆRS with its estimated value Λ. The resulting MLE is denoted by ˜RS and given as

˜RS =

m2∑i=1

(λx,i − 1)+ux,iuHx,i (3.35)

where λx,i, ux,i, ∀ i = 1, . . . , M are the eigenvalues in descending order and the corre-sponding eigenvectors of the matrixes

˜Rx = Λ−1/2RxΛ−1/2 (3.36)

respectively. Assuming r2 is the largest index of λx,i which satisfies λx,i ≥ 1, thenm2 = min(r, r2) holds.

Finally, the test statistic of the first GLRT-based method called GLRT1 is calculatedby inserting the MLEs into the general form (3.3):

TGLRT1 = ln p(x|H1)− ln p(x|H0) = N

( m2∑i=1

λx,i − lnm2∏i=1

λx,i −m2

). (3.37)

3.2.2 Known Noise Variances

In this case, the LLFs under both hypotheses are given in (3.11) and (3.13). The onlydifference to Section 3.2.1 is that the noise covariance matrix Λ is now known atthe SUs. Thus, under H0, there is no unknown parameter; under H1, only the signalcovariance matrix RS is unknown. Its estimation underH1 is given in (3.18) expressedwith the perfect noise information Λ. Taking the MLE of RS into the GLRT detector(3.3) results in the test statistic of the second GLRT-based method termed GLRT2

TGLRT2 = N

( m1∑i=1

λx,i − lnm1∏i=1

λx,i −m1

). (3.38)

where m1 was given in Lemma. 3.2.1 that m1 = min(r, r1).Remark: The results of the proposed GLRT-based methods include some special

cases introduced in the literature.

• Considering that Λ has the same diagonal entries, i.e., the noise variances arethe same for each SU. If RS is full rank, the test statistic of GLRT1 reduces to thesphericity test in [92] with unknown Λ, while the test statistic of GLRT2 reducesto the signal-subspace eigenvalue test in [148]. If the rank of the signal space isequal to one, both test statistics match the results in [132] and [122].

• Considering that Λ has different diagonal entries, i.e., the noise variances arenot identical for each SU, and assuming Λ is unknown, if the rank of the signalspace is one, the test statistic of GLRT1 is the same as the λ1 test in [84].


Therefore, the proposed GLRT methods provide a general form of a binary hypothesistesting problem in which the noise covariance matrix is a diagonal matrix and thesignal space is unknown but with available rank information.

3.2.3 Performance Evaluation

To evaluate the algorithm performance, we consider a CR network with M = 10 SUsdetecting one PU over N = 400 samples. The SNRs at the SUs are equal to [-8, -10,-16, -12, -8, -10, -11, -14, -10, -12] dB, respectively. The noise uncertainty effect of thesensing links is evaluated. Specifically, the estimated noise variances at each SU are1, 0.871, 1.15, 0.871, 1, 0.871, 1.15, 0.871, 0.871, and 1.15, respectively. The actual noisevariance is σ2

m = σ2m/α, where 10 log10 α is uniformly distributed in the region [−B, B]

with the noise uncertainty factor B in dB [113]. For example, B = 0 dB indicates thatthe noise variance is precisely known.

Five sensing methods are compared:

• ED: The test statistic is the sum of the signal energy received at all SUs

TED = NRx. (3.39)

• GLRT1: The test statistic is given in (3.37).


• Sphe.: The sphericity test in [92] assuming the noise variances are the same foreach link. The test statistic is

TSphe. =1M∑M

i=1 λx,i∏Mi=1(λx,i)1/M . (3.40)

• λ1 test: The test statistic can be similarly obtained as in [84] assuming the rankof the primary signal space is one

Tλ1 =λx,1

1M∑M

i=1 λx,i. (3.41)

• Rank ref.: The closed-form test statistic in [108] assuming the rank of the primarysignal space is known and noise variances are unknown

TRank ref. =

∏r

i=1 [Rx]i,idet Rx

, r ≥ M−√

M∑ri=1(λx,i − ln λx,i

), otherwise.

(3.42)

The algorithm is also based on the GLRT principle2.2 This work [108] is published after our conference paper [46] relevant to this section. Both works

investigated the similar problem and shared some similarity in the derivation.


10−3 10−2 10−1 1000

0.2

0.4

0.6

0.8

1

PFA

P D

B = 0 dB

GLRT2EDGLRT1Rank ref.λ1 testSphe.

10−3 10−2 10−1 1000

0.2

0.4

0.6

0.8

1

PFA

P D

B = 1 dB

Figure 3.1: ROC curve for different noise uncertainty parameters B.

10−3 10−2 10−1 1000

0.2

0.4

0.6

0.8

1

PFA

P D

B = 0 dB

GLRT2EDGLRT1Rank ref.λ1 testSphe.

10−3 10−2 10−1 1000

0.2

0.4

0.6

0.8

1

Probability of false alarm

Prob

abili

tyof

dete

ctio

n

B = 1 dB

Figure 3.2: ROC curve for different noise uncertainty parameters B.

Figure 3.1 plots the ROC curves, i.e., the detection probability PD vs. the falsealarm rate PFA for different noise uncertainty parameters B. We assume there arer = 2 independent transmit streams sent by the PU. This satisfies the conditionr < M −

√M. The number of sensing samples is N = 200. On the one hand,

concerning the subplot on the left hand side, the noise variances are perfect known,i.e., B = 0 dB. It is shown that the GLRT2 method outperforms other methods dueto the exploitation of a priori information of noise variances and rank. Without theknowledge of the noise variances, the ED outperforms the other methods, but it issensitive to the noise uncertainty as shown later. For the other four sensing methodsthat the noise variances have been estimated, the proposed GLRT1 and the Rank ref.algorithm show the best performance since they use the exact rank information to ex-


tract the useful signal subspace from the received signal. Besides, GLRT1, the Rank ref.algorithm, and λ1 test outperform the sphericity test due to the consideration of theunbalanced noises of multiple SUs. On the other hand, the subplot on the right handside depicts the ROC curves for the noise uncertainty case with B = 1 dB. Comparedto the left hand side figure, it is observed that the GLRT1, the Rank ref. algorithm,the λ1 test, and the sphericity test are insensitive to such effect, since they estimatethe noise variances before applying the GLRT principle. GLRT1 and the Rank ref. al-gorithm outperform other methods because it additionally considers the exact rankinformation of the signal subspace. Moreover, the proposed GLRT1 and the Rank ref.algorithm have the comparably same performance.

Next, we compared the ROC curves of different sensing algorithms in Figure3.2 under the condition that r ≥ M −

√M. Here we allow for the PUs to transmit

r = 8 independent streams. The number of sensing samples is N = 600. Similarobservations are concluded as in Figure 3.1 except that the Rank ref. algorithm issuffered from a slight performance loss compared to GLRT1 algorithm. The reasonis explained as follow. Under the condition that r ≥ M−

√M, the work [108] stated

that the primary signal space has no further special structure besides positive definiteHermitian property; whereas we additionally use the rank information to constructthe primary signal space under such circumstances. Thus, the MLE of the unknownparameters here is more accurate, yielding better ROC performance of the GLRT-based methods.

3.3 Spectrum Sensing Under Fading Channels

Cooperative spectrum sensing exploit the multiuser diversity to improve the sensingsensitivity and reliability. However, the performance gain over the local sensing issubject to the fading effect of both the sensing and reporting channels [77]. In order toaddress this issue, the SUs need to jointly make the sensing decision while exploringthe known CSI of the network.

In this section, we study the soft-decision fusion rules for the cooperative sens-ing. The performance here can be considered as the upper bound achieved by thequantization-based fusion. Both cases, where the sensing and the reporting channelsare assumed to be either slow or fast fading channels, are considered. For slow fadingchannels, the CSI refers to the channel information of block fading channels, while forfast fading channels, the CSI refers to the statistical CSI parameters. We focus on thestudy in a practical scenario where only the CSI and noise variances of the reportingchannels are available. This assumption is justified since the SUs may transmit pilotsto the FC for the estimation of the CSI of the reporting channels, while the informationof the sensing channels are difficult to obtain due to the lack of cooperation betweenthe PUs and the SUs.

Figure 3.3 depicts a cooperative CR network composed of K SUs and a secondaryFC. The cooperative sensing includes two parts: sensing and reporting. In the sensingprocedure, each SU performs spectrum sensing based on the observation over N sam-

3.3. Spectrum Sensing Under Fading Channels 33

Figure 3.3: Cooperative sensing in the secondary network with K SUs and FC attime instant n.

pling times. Under H1, the primary signal is assumed to be a zero-mean temporallyand spatially white signal with unit power for each transmit antenna. The primarytransmit signal at the nth time instant is s[n] ∈ CNt×1 where Nt is the number of trans-mit antennas at the PU. Assuming each SU has a single receive antenna and denotinghk[n] ∈ C1×Nt as the stationary Rayleigh fading channel from the PU to the kth SUat the nth instant, we obtain the matrix H[n] =

[h1[n]T, . . . , hK[n]T

]T . The signal partreceived by the SUs at the nth instant is

s[n] = H[n]s[n] (3.43)

and ΣS is the covariance matrix of s[n]. Assuming that xk[n] is the nth receivedsignal sample at the kth SU, the received signals at K SUs are stacked into a vectorx[n] = [x1[n], . . . , xK[n]]

T as

x[n] =

{w[n], H0

s[n] + w[n], H1, ∀ n = 1, . . . , N (3.44)

where the temporally and spatially white noise is w[n] = [w1[n], . . . , wK[n]]T with

nk[n], ∀ k = 1, . . . , K being the noise sample for the kth SU at the nth time instant. Weassume w[n] ∼ CN (0, σ2

wIK). The signal samples {s[n]}Nn=1 and the noise samples

{w[n]}Nn=1 are mutually independent.

In the reporting procedure, the FC receives the signals relayed by each SU throughthe reporting channels in an orthogonal manner to exploit the spatial diversity [106]

y[n] = G[n]x[n] + v[n] (3.45)


where G = diag{g[n]} with

g[n] = [g1[n], . . . , gK[n]]T

and gk[n] representing the fading coefficient from the kth SU to the FC. We remarkthat gk[n] is not restricted to Rayleigh fading channels, e.g., a more general Ricianfading channel model can be applied if there is a line of sight between the SU andthe FC. v[n] = [v1[n], . . . , vK[n]]T is the vector containing the temporally and spatiallywhite noise samples of the reporting channel and v[n] ∼ CN (0, σ2

v IK).Two kinds of fading models are applied to sensing and reporting channels. We

also assume different types of available CSI for two channel models, respectively.

• Fast fading channels: The instantaneous CSI varies for different time instantsand is hard to obtain. Thus, only statistical CSI is assumed to be available. Weuse ΣH = E{H[n]H[n]H}, g[n] ∼ CN (g, Σg). Consequently, we obtain

ΣS = ΣH, Rg = Σg + ggH (3.46)

where Rg is the correlation matrix of g[n].

• Slow fading channels: The CSI {H[n]} and {g[n]} are the block fading valuesthat are fixed for ∀ n = 1, . . . , N. Therefore, the time argument n in H[n] andg[n] is dropped for simplicity. We obtain

ΣS = HHH, Rg = ggH. (3.47)

We aim at designing the sensing rule at the FC to combine y[n], ∀ n = 1, . . . , N.According to the Neyman-Pearson optimality criterion, for a binary hypothesis testingproblem, the optimal test statistic to maximize the PD given a fixed PFA is the followinglog LRT (LLRT) scheme [66]:

ΛLLRT(y) = lnp(y|H1)

p(y|H0)

H1≷H0

γ (3.48)

where γ is the threshold and

y =[yT(1), . . . , yT(N)

]T.

The conditional PDF p(y|Hi), i = 0, 1 is difficult to calculate since each y[n] contains,e.g., a multiplication of two Gaussian distributions [53]. Instead, we approximatep(y|Hi) by using a Gaussian distribution pG(y|Hi). Since the observations {y[n]}N

n=1are independent with the Gaussian assumption, we have

pG(y|Hi) =∏N

n=1pG(y[n]|Hi).


Dropping the time argument n for y[n], we obtain

pG(y|Hi) ∼ CN (µi, Σi)

where µi and Σi are

µi = E{y|Hi} (3.49)

Σi = E{yyH|Hi} − µiµHi . (3.50)

The closed-form expressions of (3.49) and (3.50) are calculated in Appendix A.1. Theresults are

[µi]k = gk xk,i (3.51)

[Σi]k,k′ = σ2v δk,k′ + E(xkx∗k′)E(gkg∗k′)− [µi]k[µi]

∗k′

(3.52)

with

gk = E(gk)

xk,i = E(xk|Hi).

According to (3.44), the received signal x can be approximated by the followingGaussian distribution

x ∼{CN (0, σ2

wIK), H0

CN (0, ΣS + σ2wIK), H1.

(3.53)

Integrating (3.53) into (3.51) and (3.52), the mean and covariance matrices of y areobtained as

µ0 = 0, Σ0 = σ2wIK � Rg + σ2

v IK (3.54)

µ1 = 0, Σ1 = (ΣS + σ2wIK)� Rg + σ2

v IK (3.55)

where ΣS and Rg are given in (3.46) and (3.47) for two kinds of fading channels,respectively.

3.3.1 Approximated Likelihood Ratio Test

Due to the lack of the analytical form of the exact PDF p(y|Hi), i = 1, 2, we replaceit with the Gaussian approximation pG(y|Hi). The resulting near-optimal method isnamed approximated likelihood ratio test (ALRT) method under the assumptions ofthe known CSI and noise variances of both sensing and reporting channels.

In particular, substituting pG(y|Hi) into (3.48) and integrating the constant itemsinto the threshold design, the test statistic of the ALRT detector is given as

ΛALRT(y) =N∑

n=1

yH[n](

Σ−10 − Σ−1

1

)y[n]. (3.56)


3.3.2 Energy Detection

If the FC sums up the energy of the signals relayed by each SU, the detection schemebecomes the widely-used ED

ΛED(y) =N∑

n=1

yH[n]y[n]. (3.57)

The test statistic ΛED can be considered as the special case of ΛALRT when(

Σ−10 − Σ−1

1

)is equal to an identity matrix. The ED only requires knowledge of the primary signalsuch as the center frequency and bandwidth, but neither a priori information of thesignal structure nor of the CSI.

3.3.3 GLRT-Based Detection

Given the approximated signal model, we apply the GLRT principle (3.3) and yield

ΛGLRT(y) = lnpG(y|H1, θ1)

pG(y|H0, θ0)(3.58)

with

θ0 = arg maxθ0

pG(y|H0, θ0) (3.59)

θ1 = arg maxθ1

pG(y|H1, θ1). (3.60)

Note that the exact distribution p(y|Hi) with i = 0, 1 is replaced with the approxi-mated Gaussian distribution pG(y|Hi) for the sake of computational tractability.

3.3.3.1 Unknown Noise Variances

In this case, both noise variance σ2w of the sensing channels and ΣS need to be esti-

mated under both hypotheses H0 and H1.First, the LLF under H0 with the unknown σ2

w is written as

ln pG(y|H0) =− KN ln π − N ln det Σ0 − Ntr{

Σ−10 Ry

}. (3.61)

where

Ry =1N

N∑n=1

y[n]yH[n](a)= UyΛyUH

y . (3.62)

The step (a) in (3.62) indicates the eigenvalue decomposition of Ry. The term

Λy = diag{

λ1,y, . . . , λK,y}


contains all the eigenvalues of Ry as diagonal entries in descending order. After somemathematical manipulation, the MLE of σ2

w is derived as

σ2w|H0 =

1K

tr{C} (3.63)

with

[C]i,j =

[Uydiag

{(λ1,y − σ2

v)+ , . . . ,

(λK,y − σ2

v)+}UH

y

]i,j[

Rg]

i,j

Similarly, the LLF under H1 with the unknown ΣS and σ2w is

ln pG(y|H1) =− KN ln π − N ln det Σ1 − Ntr{

Σ−11 Ry

}. (3.64)

The MLE of the unknown parameters under H1 is

ΣS + σ2wIK|H1 = C. (3.65)

Integrating (3.63) and (3.65) into (3.61) and (3.64), the GLRT test statistic is given as

ΛGLRT3(y) =− N lnm3∏

k=1

λk,y

σ2v− Nm3 − N

K∑k=m3+1

λk,y

σ2v

(3.66)

+ N lnK∏

k=1

λk,e + Ntr{

E−1Ry

}(3.67)

whereE =

(σ2

w|H0

)IK � Rg + σ2

v IK

and λk,e, ∀ k = 1, . . . , K, are the eigenvalues of E. The value m3 refers to the largest m3such that

λm3,y ≥ σ2v .

3.3.3.2 Known Noise Variances

If the covariance matrix ΣS is unknown, i.e., the power of the primary signal andthe CSI of the sensing channel are unknown, they can be estimated by exploiting thestructure of the sample covariance matrix of the received signals. Specifically, the LLFunder H0 is

ln pG(y|H0) = −KN ln π − N ln det Σ0 −N∑

n=1

yH[n]Σ−10 y[n] (3.68)


where all parameters in (3.68) are known, i.e., there is no parameter to be estimatedin (3.58). By using a decomposition of the form

Σ0 = LH1 L1

(3.68) is reformulated as

ln pG(y|H0) = −KN ln π − N ln det LH1 L1 − Ntr

{Ry}

(3.69)

where

Ry = L−H1

(1N

N∑n=1

y[n]yH[n]

)L−1

1 .

Similarly, the LLF under H1 is reformulated as follows due to the temporal white-ness of s[n], ∀ n = 1, . . . , N

ln pG(y|H1) = −KN ln π + N ln det(

L−11 BL−H

1

)− Ntr

{RyB

}. (3.70)

where

B =(

L−H1(ΣS � Rg

)L−1

1 + IK

)−1. (3.71)

According to (3.71), the unknown parameter ΣS is only included in B, i.e., the un-known parameter θ1 in (3.58) is B. The MLE of B is obtained by solving the followingconstrained optimization problem

maxB

ln det B− tr{

RyB}

s.t. 0 � B � IK.

Similar to [148, Appendix], we obtain the MLE of B

B = Uydiag{

min(

λ−11,y , 1

), . . . , min

(λ−1

K,y, 1)}

UHy (3.72)

where Uy contains the eigenvectors of Ry and λk,y, ∀ k = 1, . . . , K are the correspond-ing K eigenvalues listed in descending order, i.e., λ1,y ≥ λ2,y ≥ . . . ≥ λK,y.

Taking the MLE result (3.72) into (3.70) and subtracting (3.69), it results in the teststatistic of GLRT4

ΛGLRT4(y) = N ln det B− Ntr{

RyB}+ Ntr

{Ry}

. (3.73)

Using (3.72), we reformulate the two terms containing B as

ln det B = − lnm4∏

k=1

λk,y (3.74)


tr{

RyB}= m4 +

K∑k=m4+1

λk,y (3.75)

where m4 refers to the largest m4 such that λm4,y ≥ 1. Integrating (3.74) and (3.75) into(3.73) results in the final test statistic of GLRT4

ΛGLRT4(y) = Nm4

1m4

m4∑k=1

λk,y − ln

( m4∏k=1

λk,y

) 1m4

− 1

. (3.76)

Remark: The performance of the GLRT-based methods depends on the distinguishablestatistical properties of the estimated parameters under both hypotheses. Specifically,the following two properties are exploited to differentiate binary hypotheses : underthe hypothesis H1, the received signals from multiple SUs at the FC are correlatedand have unbalanced power levels due to different properties of the sensing and thereporting channels; on the contrary, under the hypothesis H0, the received noises areuncorrelated and their variances are same. However, in fast fading scenario, if thereporting channel has low spatial correlation, i.e., Rg is a scaled identity matrix, thecorrelation effect diminishes at the FC according to (3.54) and (3.55). This resultsin a performance degradation of the GLRT-based methods, since only the property ofunbalanced power levels is used to identify the presence and absence of the PU signal.Nevertheless, the performance can be improved by other approaches, e.g., prolongingthe sensing length.

3.3.4 Performance Evaluation

To numerically evaluate the performance, we consider a CR network with K = 4 SUsdetecting one PU with a single transmit antenna. The primary signals are quadraturephase-shift keying modulated signals with unit variance. Each channel hk[n] or gk[n]is assumed to be a Rayleigh fading channel. The SNRs of the sensing and the reportingchannels are specified in each figure. The noise variance of the reporting links isσ2

v = 0.8. The noise uncertainty effect of the sensing links is considered, i.e., theestimated noise variance is σ2

w = 1.5 and the true noise variance is σ2w = σ2

w/α, where10 log10 α is uniformly distributed in [−B, B] with noise uncertainty factor B in dB[113]. For example, B = 0 dB indicates that the noise variance is precisely known.

We compare the following four sensing algorithms.

• ALRT: The test statistic is given in (3.56).

• ED: The test statistic is given in (3.57).



Figure 3.4 plots the ROC curves for slow fading channels. The observation lengthis M = 1000. Two cases of noise uncertainty are considered: B = 0 dB and B = 1 dB.


10−3 10−2 10−1 1000

0.2

0.4

0.6

0.8

1

PFA

P D

B = 0 dB

ALRTGLRT4EDGLRT3

10−3 10−2 10−1 1000

0.2

0.4

0.6

0.8

1

PFA

P D

B = 1 dB

Figure 3.4: ROC curve with noise uncertainty for slow fading channels. The SNRsof the sensing links are set to [-18, -11, -16, -10] dB and the SNRs of thereporting links are [8, 10, 12, 6] dB, M=1000.

10−3 10−2 10−1 1000

0.2

0.4

0.6

0.8

1

PFA

P D

B = 0 dB

ALRTGLRT4EDGLRT3

10−3 10−2 10−1 1000

0.2

0.4

0.6

0.8

1

PFA

P D

B = 1 dB

Figure 3.5: ROC curve with noise uncertainty for fast fading channels. The SNRsof the sensing links are set to [-18, -11, -16, -10] dB and the SNRs of thereporting links are [8, 10, 12, 6] dB, M=3000.

In the legend, the value of B is denoted in parentheses. We observe that if the noisevariance σ2

w is perfectly known, the ALRT method performs the best due to completea priori knowledge. Without the knowledge of ΣS, GLRT4 uses the hidden knowl-edge in the received signals and outperforms the ED. If a noise uncertainty exists,GLRT3 always performs the best in the region of interest since the noise variance isestimated. As expected, GLRT3 is shown to be insensitive to the uncertainty, while

3.4. Summary 41

the performance of the other three methods degrades severely. Figure 3.5 depicts theROC curves for fast fading channels. The corresponding correlation matrix ΣH or Σg

is a Toeplitz matrix [ΣH]i,j = 0.1|i−j| and [Σg]i,j = 0.1|i−j|, respectively. The observationlength is M = 3000. Similar observations are obtained as in Figure 3.4.

3.4 Summary

In CR networks, cooperative spectrum sensing is more favorable than local spectrumsensing due to its improved sensitivity and the effectiveness in avoiding the hiddenterminal problem. However, the performance of cooperative sensing is subject to sev-eral realistic restrictions. For example, it is difficult for the SUs to acquire perfectknowledge of the primary signal space and the CSI of the entire network. Further-more, the fading of the sensing and reporting channels can deteriorate the sensingperformance. Hence, we have investigated effective spectrum sensing methods aim-ing at tackling the aforementioned challenges.

The spectrum sensing problems with limited a priori knowledge can be modeledas binary hypothesis testing problems with unknown parameters. In this chapter, wehave applied the GLRT principle to two realistic sensing examples. Firstly, assumingan unknown structure of the primary signal space at the SUs, we exploited the rankinformation to extract its structure. The GLRT-based sensing methods were then de-signed corresponding to two scenarios, i.e., unknown or known noise variances. Theresulting algorithms also generalize some special cases in the literature. Secondly, weconsidered the influence of the fading sensing and reporting channels on cooperativesensing. The SUs are assumed to only have partial CSI of the channels but withoutknowing the structure of the primary signal space. We derived the sensing algorithmsunder two scenarios, i.e., the noise variances of the sensing channels are unknown orknown. Overall, the proposed GLRT-based algorithms effectively utilize the avail-able information at the SUs in countering the performance degradation caused byenvironment uncertainties, e.g., fading effects and noise uncertainty.

Chapter 4

Power Allocation with Partial PrimaryCSI

The underlay paradigm allows for the simultaneous transmission of PUs and SUs.Due to the hierarchical usage of the resources, the main goal is to design the secondarytransceiver strategies in order to constrain the performance degradation caused to thePUs below some tolerable limit.

Power allocation is an effective way for interference mitigation. Several works[8, 33, 37, 64, 120] have considered the design of power control strategies using theconventional metric to evaluate the performance degradation of the PU, i.e., the ITconstraint denoting the received interference power at the PR. Alternatively, limitingthe primary capacity loss is the ultimate goal in regulating the secondary transmis-sion. To this end, the authors in [65, 149] demonstrated that a gain on the secondaryrate is achieved by further exploiting the primary CSI instead of considering solelythe IT constraint during the system design.

In practice, the SUs can only acquire partial CSI of the entire network, especiallyregarding the CSI related to the PUs. Two kinds of partial CSI models were consideredin the literature: outdated CSI [73, 89, 94, 110, 121] and statistical CSI [12, 28, 80, 116].On the one hand, outdated CSI is obtained on a real-time basis subject to the delaycaused in channel feedback or estimation. The works [73, 89] found that the lowcorrelation between the outdated and the actual CSI yields a large performance loss ofthe secondary transmission. On the other hand, statistical CSI changes on a time scalethat is much larger than the channel coherence time. Given such CSI, the performanceloss of the PUs can be limited in an average or a probability-constrained manner.

Given the power control strategies, it is desirable to obtain insights on the achiev-able performance of the SUs at the expense of the performance degradation to thePUs. Such performance analysis can be used for a performance assessment and toprovide guidelines in selecting system parameters for power adaptation strategies.However, the optimal power strategy is usually not given in closed form. Thus, itbrings in mathematical challenges in the performance analysis. An alternative wayis to develop a near-optimal strategy. If the corresponding analytical performance isobtained, it can be a good approximation of the optimal achievable performance.

In this chapter, power allocation strategies for the secondary transmission areinvestigated subject to different QoS constraints on the primary link under Rayleighfading channels assuming only partial CSI related to the PR is available at the ST.After introducing the system model in Section 4.1, we study the power allocation inSection 4.2 subject to an average IT constraint and a transmit power constraint on theST. Motivated by the benefit of exploiting the primary CSI [149], the power allocation

43

44 Chapter 4. Power Allocation with Partial Primary CSI

Figure 4.1: Spectrum sharing system with a primary and a secondary link.

strategies are designed in Section 4.3 using the statistical CSI of the primary linksubject to an outage probability constraint on the PU instead of an IT constraint. Weaim at not only deriving the optimal power allocation strategies but also designing thelow-complexity near-optimal strategies with the corresponding performance analysis.

The results presented in this chapter have been published in part by the authorin [40, 41, 47–49]1.

4.1 System Model

Figure 4.1 depicts the considered cross-interfering spectrum sharing system in whicha secondary link coexists with a primary link. We assume that a single antenna isequipped at the ST, the SR, the PT, and the PR, respectively. The transmit symbols ofthe PT and the ST at the kth time instant are given by x1[k] ∼ CN (0, 1) and x2[k] ∼CN (0, 1), respectively. We assume that the PT uses non-adaptive power transmissionwith the power denoted by P1, while the ST optimizes the power P2[k] according to theCSI at the kth time instant. We remark that similar to [37, 69], here the term “power”refers to the instantaneous power averaged over the transmit symbols.

All channels are assumed to be stationary, ergodic, and mutually independentRayleigh flat fading channels in a slow-fading scenario. Instantaneous CSI of the PT-PR link, the PT-SR link, the ST-PR link, and the ST-SR link is given by h11[k], h12[k],h21[k], and h22[k], respectively. The noise of the primary and the secondary link isgiven by np[k] ∼ CN (0, σ2

P) and ns[k] ∼ CN (0, σ2S), respectively, with positive and

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4.2. Power Control with Interference Power Constraint 45

finite σ2P and σ2

S. Consequently, the received signals at the PR and the SR at the kthtime instant are denoted by y1[k] and y2[k], respectively:

y1[k] =√

P1h11[k]x1[k] +√

P2[k]h21[k]x2[k] + np[k] (4.1)

y2[k] =√

P2[k]h22[k]x2[k] +√

P1h12[k]x1[k] + ns[k]. (4.2)

The channel power gain is denoted by

gij[k] =∣∣hij[k]

∣∣2 , i, j = 1, 2 (4.3)

which is exponentially distributed with the PDF

fgij(x) =

{lije−lijx, x ≥ 0

0, x < 0i, j = 1, 2 (4.4)

with lij indicating the rate parameter. In practice, the instantaneous CSI g11[k] andg21[k] is difficult to obtain at the ST due to limited cooperation between the SU andthe PU. However, we assume statistical parameters l11 and l21 to be available at the STdue to the exploitation of side information [61]. For the PT-SR link, either statisticalCSI parameter l12 or the the instantaneous CSI g12[k] is available at the ST, dependingon the scenario whether the ST can obtain statistical parameters by the location infor-mation or obtain the instantaneous CSI via estimation or feedback. Additionally, weassume both the ST and the SR know perfect instantaneous CSI of the secondary linkg22[k] and the SR additionally knows the instantaneous CSI of the PT-SR link g12[k].For the remainder of the chapter, we omit the time argument k for simplicity.

We consider two kinds of QoS constraints to restrict the performance degradationof the primary transmission: average IT constraint and outage probability constraints.Besides, the peak power of the ST is limited. The power control strategies aimingat maximizing the achievable rate of the secondary link subject to aforementionedconstraints are investigated in the remainder of this chapter.

4.2 Power Control with Interference Power Constraint

4.2.1 Known CSI of the PT-SR Link

Recalling the system model in Figure 4.1, we assume the ST and the SR know instan-taneous CSI g12 and g22, but only statistical CSI parameter l21. This assumption isjustified due to the fact that such instantaneous CSI can be possibly obtained in twoways: First, the SR can estimate g12 and g22 by exploiting the pilot symbols from theST and the PT and then feedback it to the ST; second, the ST can also estimate suchCSI by exploiting the reciprocity in the TDD transmission.

Two power constraints are considered. First, the average interference power con-straint PI is imposed concerning the protection of the PU. Second, the peak power con-


straint PS is set for the ST. We assume Gaussian input signallings and single-user de-tection at the SR in which the interfering signal from the PT is taken as noise [17,126],The optimal power strategy subject to both power constraints is obtained by solvingthe following optimization problem

maxP2(g12, g22)

Eg12, g22 ln

(1 +

P2(g12, g22)g22

P1g12 + σ2S

)(4.5)

s.t. P2(g12, g22) ≤ PS, ∀ g12 ≥ 0, g22 ≥ 0P2(g12, g22) ≥ 0, ∀ g12 ≥ 0, g22 ≥ 0Eg12, g22, g21 {P2(g12, g22)g21} ≤ PI

where the notation P2(g12, g22) is used because the power is adapted to the instanta-neous knowledge g12 and g22. The unit of the objective function is nats/s/Hz.

Since the ST only has statistical CSI of the ST-PR link, the optimal power allocationstrategy is independent from each instantaneous channel realization g21. Consideringthe mutual independence of the CSI of each link, we convert the third constraint of(4.5) to the following inequality

Eg12, g22 {P2(g12, g22)} ≤ PI l21. (4.6)

For the sake of simplicity, introducing the variable t as

t =σ2

S + P1g12

g22(4.7)

we represent P2(g12, g22) as P2(t) and reformulate the optimization problem (4.5) as

maxP2(t)

R (P2(t)) (4.8)

s.t. P2(t) ≤ PS, ∀ t > 0P2(t) ≥ 0, ∀ t > 0Et {P2(t)} ≤ PI l21

where R (·) is defined as

R (x(t)) = Et

{ln(

1 +x(t)

t

)}. (4.9)

Note that since the optimization variable P2(t) in (4.8) is a function instead of a vectorin Euclidean space, the optimization theory in function space [58, 85] is applied laterto derive the optimal solution. A brief revision of KKT conditions for functionaloptimization is given in [87].


4.2.1.1 Optimal Power Allocation

The optimization problem (4.8) is a constrained convex problem and the Slater con-dition is satisfied. Then the KKT conditions are sufficient and necessary conditions,i.e., the KKT solution is the global optimal solution [111]. This argument provides thetheoretical basis to design the optimization strategy.

Proposition 4.2.1. The KKT solution to the optimization problem (4.8) is

P?2 (t) =

0, t ≥ v?

v? − t, v? − PS < t < v?

PS, 0 ≤ t ≤ v? − PS.(4.10)

The non-negative parameter v? is the inverse of an optimal Lagrangian multiplier λ? relatedto the constraint (4.6). It is chosen to fulfill the average power constraint (4.6).

Proof. The Lagrangian of (4.8) is written as

L = Et

{ln(

1 +P2(t)

t

)}+ λ (Et {P2(t)} − PI l21) (4.11)

where the non-negative Lagrangian multiplier λ corresponds to the constraint (4.6).Herein, no Lagrange multiplier is assigned to the peak power constraints since theyare considered in the generalized KKT conditions as shown later.

According to the generalized KKT conditions for functional optimization [85, 87],if the optimal solution P?

2 (t) is a regular point, it satisfies the following conditions:

dldP2(t)

∣∣∣∣∣P2(t)=P?

2 (t)

= 0, 0 < P?

2 (t) < PS

≤ 0, P?2 (t) = 0

≥ 0, P?2 (t) = PS

(4.12)

λ? ≥ 0 (4.13)λ? (Et {P?

2 (t)} − PI l21) = 0 (4.14)Et {P?

2 (t)} ≤ PI l21 (4.15)

where the function l is defined as

l = ln(

1 +P2(t)

t

)+ λ? (P2(t)− PI l21) (4.16)

and the derivative of (4.12) with respect to (w.r.t.) P2(t) is

dldP2(t)

=1

t + P2(t)− λ?. (4.17)

By solving the KKT conditions (4.12), (4.13), and using v = 1/λ, we obtain the solutionas given in (4.10). The KKT conditions (4.14) and (4.15) are used to determine v?.Particularly, if the constraint (4.6) is inactive at the optimal solution, v? should be


infinity since λ? = 0 according to (4.14). Under this circumstance, the optimal solutionis P?

2 (t) = PS for all t, which leads to Et {P?2 (t)} = PS < PI l21. If PS = PI l21, it is easy

to see that the optimal solution is also P?2 (t) = PS, thus the calculation of v? is not

required. Thus, we only focus on calculating v? for PS > PI l21. In this case, v? shouldbe chosen to let the equality Et {P?

2 (t)} = PI l21 hold.

In order to determine three regions of t to divide the power allocation strategy in(4.10), it is essential to efficiently calculate v?. As discussed, we only need to obtainv? for PS > PI l21. Defining

Qavg(v) = PS

∫ v−PS

0f (t)dt +

∫ v

v−PS

(v− t) f (t)dt (4.18)

where f (t) is the PDF of t. We need to choose v? to let

Qavg(v?) = PI l21 (4.19)

hold under such circumstance. Taking the derivative of Qavg(v) over v results in

Q′avg(v) =

dQavg(v)dv

=

∫ v

v−PS

f (t)dt > 0 (4.20)

which shows that Qavg(v) is a strictly monotonical increasing function on v, ∀ v > 0.This property makes some general numerical approaches, i.e., the bisection method,feasible to seek for v?. However, due to the lack of the explicit expressions of theseintegrals in (4.18), we need to evaluate v in each bisection step over a large number ofchannel realizations, which is very time consuming.

We propose an efficient method to determine v? that uses the closed-form expres-sions of Qavg(v) and its derivative Q

′avg(v). Specifically, using the Newton’s method,

we calculate the following iteration until a sufficiently accurate value is reached

v(n+1) = v(n) −Qavg(v(n))− PI l21

Q′avg(v(n))(4.21)

where the closed-form expressions of Qavg(v) and Q′avg(v) are given in Appendix A.2.

The initial point of the iteration need to be selected carefully to guarantee the methodconverges [105].

The overall method seeking for v? is as follows. We first choose some initial valueof v and apply the Newton’s method in (4.21). If it fails to converge after a pre-definednumber of iterations, it means that the initial point is not properly chosen. In suchcases, an alternative bisection method is applied by using the analytical expression ofQavg(v) in each iteration step.

Remarks: If the interference of the PT-SR link is ignored, then the optimizationproblem can be solved similarly to the method in [69]. The resulting power alloca-tion strategies are also divided into three regions. Herein, we not only additionallyconsider this interference term, but also derive the analytical forms of Qavg(v) and


Q′avg(v) in (A.6) and (A.7), respectively. Based on these closed-form expressions, an

efficient method is proposed to calculate the threshold value v in determining differ-ent power allocation regions.

4.2.1.2 Performance Analysis

We characterize the the achievable performance of a single-antenna secondary linkusing the optimal power allocation strategy in (4.10).

Proposition 4.2.2. The achievable performance of exploiting the power allocation strategy(4.10) is given as

• If PS ≤ PI l21, then

R (P?2 (t)) = lim

v?→∞r (v? − PS, 1, PS) (4.22)

• If PS > PI l21, then

R (P?2 (t)) = r (v? − PS, 1, PS) + r (v?, 0, v?)− r (v? − PS, 0, v?) (4.23)

where the function r(T, α, β) with α + β/T > 0. For T ≤ 0, r(T, α, β) = 0; for T > 0,

r(T, α, β) =

bg

(ln(α + γ)

γ + he−gγ +

1α− h

(eghE1(g(h + γ))− egαE1(g(α + γ))

)), h 6= α

bg

(ln(γ + α)

γ + αe−gγ +

e−gγ

γ + α− gegαE1(g(γ + α))

), h = α

(4.24)

In above, we use the variables

γ =β

T, b =

l12σ2S

P1, g =

l22σ2S

β, h =

l12β

l22P1. (4.25)

Proof. See Appendix A.3.

Two remarks are addressed here and will be verified in the numerical results.

1. We consider the relation between the achievable rate and the peak transmitpower constraint PS given a certain value of PI .

First, when PS increases from zero and fulfills PS < PI l21, v? is infinite. It isobvious that the derivative of (A.9) over PS is larger than zero. Therefore, theachievable rate increases monotonically with PS.


Second, we consider PS increasing beyond PI l21 but smaller than PS,1 (its valuewill be discussed later). Calculating the derivative of Qavg in (A.6) over PS,2 wehave

dQavg

dPS=

∫ v−PS

0f (t)dt ≥ 0. (4.26)

Combining it with (4.20) means that an increasing PS yields a decreasing v? sinceQavg is fixed in this case.

Third, keeping PS increasing and consequently yielding a decreasing v?, wereach some boundary point PS,1 beyond which v? − PS < 0 holds. Reviewing(A.9), the achievable rate remains constant even if PS increases further. TakingPS,1 = v? into (A.6), the boundary PS,1 can be calculated by solving the followingequation ∫ PS,1

0(PS,1 − t) f (t)dt = PI l21. (4.27)

2. We consider the relation between the achievable rate and the average interfer-ence constraint PI given a certain value of PS.

First, if we increase PI from zero to PI,1 = PS/l21, the mean interference con-straint should be met with equality at the optimal value, i.e., v? is finite. From(A.7) we observe that Qavg(v?) strictly increases with v?. Thus, an increasing PIyields an increasing v?. Taking the derivative of R in (A.9) with regard to v, wenote that the achievable rate increases with finite v? and PI .

Then, if PI keeps increasing and fulfills PI > PS/l21, the average interferenceconstraint is met with inequality at the optimal value in (4.8) since the optimalpower is always equal to PS. This indicates that the achievable rate is constant.

The numerical results for the achievable rate of the secondary link with the op-timal power control (4.10) are represented as follows. We set l12 = 3, l21 = 2 andl22 = 1. The noise variance of the secondary link is σ2

S = 1. We compare both thenumerical and analytical results of the optimal solution to the non-adaptive powertransmission aiming at satisfying the system constraints at every time instant:

P2(t) =

{PS, PS ≤ PI l21

PI l21, otherwise, ∀t > 0. (4.28)

Fig. 4.2 plots the achievable rate versus the transmit power constraint PS underinterference power constraints PI . The detailed description of the legend is givenbelow the figure. The analytical expression of the achievable rate matches exactlythe numerical solutions. Furthermore, an increasing PS with a fixed PI yields anincreasing achievable rate until some boundary. Increasing PS beyond this boundary

2 Herein, we take Qavg as the function of PS instead of v. Therefore, the argument (v) is omitted inQavg(v).


0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

0.5

1

1.5

2

2.5

Transmit power PS (dB)

Ergo

dic

Cap

acit

yR

(bit

s/s/

Hz) PI= 0 dB

PI= 5 dBPI= 10 dB

Figure 4.2: Achievable rate versus transmit power limit PS for different interferencepower limit PI . The lines/markers of the same color share the same pa-rameter PS. Solid lines: optimal solution (sim.); circles: optimal solutionI (analytical); dashed lines: non-adaptive power transmission (sim.).

does not affect the achievable rate. The value of the boundary can be obtained bysolving (4.27). Finally, the superiority of the optimal solution over the non-adaptivetransmission is clearly shown.

Fig. 4.3 illustrates the achievable rate versus the interference power constraint PIunder different transmit power constraints PS. The optimal solution outperforms thenon-adaptive transmission due to the adaptation to the available instantaneous CSI.We observe the rate saturates with the increasing of PI when PS is fixed. Given acertain PS, the boundary PI is given by PSl−1

21 . For example, for the case PS = 10 dB,the boundary is PI,1 = 6.9897 dB which matches the observation. When PI increasesbeyond PI,1, the optimal power allocation turns out to be a fixed power transmissionwith PS which consequently yields the constant achievable rate.

4.2.2 Unknown CSI of the PT-SR Link

Different from Section 4.2.1, we assume that the ST knows the statistical CSI of thePT-SR link instead of the instantaneous CSI. More specifically, the ST acquires theinstantaneous power gain g22 and statistical CSI l12 and l21. Two constraints are im-posed: the average interference power received at the PR is constrained by PI and thepeak power of the ST is limited by PS. The optimization problem is formulated as

R = maxP2(g22)

Eg12,g22

{ln

(1 +

P2(g22)g22

P1g12 + σ2S

)}(4.29)


0 3 6 9 12 150.5

1

1.5

2

2.5

3

3.5

4

Interference power limit PI (dB)

Ach

ieva

ble

rate

R(b

its/

s/H

z)

PS= 5 dBPS= 10 dBPS= 15 dBPS= 20 dB

Figure 4.3: Achievable rate versus interference power limit PI for different transmitpower limit PS. The lines/markers of the same color share the same pa-rameter PS. Solid lines: optimal solution (sim.); circles: optimal solutionI (analytical); dashed lines: non-adaptive power transmission (sim.).

s.t. P2(g22) ≤ PS, ∀g22 ≥ 0P2(g22) ≥ 0, ∀g22 ≥ 0Eg22, g21 {P2(g22)g21} ≤ PI .

Note that here the notation P2(g22) is used instead of P2(t) in (4.8) because the sec-ondary transmit power is only adapted to the instantaneous CSI g22. Because theexpectation operation preserves convexity [115], the objective function of (4.29) is aconcave function in P2(g22). Moreover, all the constraints in (4.29) are affine functionsof P2(g22). Hence, (4.29) is a convex optimization problem.

The average interference power constraint in (4.29) is converted into

Eg22 {P2(g22)} ≤PI

Eg21 {g21}= PI l21 (4.30)

which corresponds to an average transmit power constraint.Remarks: After reformulating the average interference power constraint into the

average power constraint, the problem (4.29) is equivalent to the problem aiming atmaximizing the secondary rate subject to both a peak and an average power con-straint. The problem formulation of (4.29) generalizes problems studied previouslyin the literature. If we set P1 = 0, (4.29) reduces to the problems considered in [69]and [71],3 i.e., without considering the interference from the PT.

3 In [71], optimal power control in a multi-antenna setup is considered as well.


4.2.2.1 Optimal Power Allocation

We can show that a regularity condition, e.g., the Slater condition, is satisfied. Usually,the optimum of such an optimization problem is mathematical tractable. However,deriving the analytical solution to (4.29) still involves several challenges, e.g., it ishard to obtain an explicit analytical form since the objective function includes anexpectation expression.

In what follows, we first reformulate the objective function as an explicit functionof the available CSI. Based on this, we solve (4.29) by applying the KKT optimalityconditions. The resulting solution requires solving a nonlinear equation. An efficientnumerical method is provided to obtain the solution and the thresholds in dividingdifferent power allocation regions.

The power gain of the interfering PT-SR link g12 follows an exponential distribu-tion with mean 1/l12. Moreover, the optimal power P2 is a function of statistical CSIl12 instead of the unknown instantaneous CSI g12. Motivated by this fact, we evaluatethe objective function of (4.29) over different channel realizations by calculating theexpectation with respect to (w.r.t.) g12. Specifically, introducing

s0 =σ2

SP1

, s1 =σ2

S + P2(g22)g22

P1(4.31)

we write the objective function as

Eg12, g22

{ln

(1 +

P2(g22)g22

σ2S + P1g12

)}

(a)= Eg22

ln

(1 +

P2(g22)g22

σ2S

)− P2(g22)g22

P1

∫ ∞

0

e−l12g12dg12

g212 +

2σ2S+P2(g22)g22

P1g12 +

σ4S+P2(g22)g22σ2

SP2

1

(b)= Eg22

{ln

(1 +

P2(g22)g22

σ2S

)−∫ ∞

0

(1

g12 + s0− 1

g12 + s1

)e−l12g12dg12

}(c)= Eg22

{ln

(1 +

P2(g22)g22

σ2S

)+ el12s1 E1(l12s1)

}− el12s0 E1(l12s0) (4.32)

where, in step (a), we use integration by parts and incorporating the PDF of g21, andin step (b) we integrating (4.31). In step (c), we use the definition of the exponentialintegral [4]

E1(x) =∫ ∞

x

e−t

tdt, R(x) > 0. (4.33)

Since (4.29) is a convex optimization problem with continuous differentiable objec-tive function and constraints and the Slater condition is satisfied, the KKT conditionsare sufficient and necessary conditions for the optimum solution [87].


Theorem 4.2.1. The optimal power allocation P?2 (g22) solving the problem (4.29) is

P?2 (g22) =

0, g22 ≤ v0(λ

?2)

p?(g22), v0(λ?2) < g22 < v1(λ

?2)

PS, otherwise.(4.34)

where λ?2 is the optimal value of the non-negative Lagrangian multiplier corresponding to the

constraint (4.30). Here, v0(λ?2) and v1(λ

?2) are the thresholds that divide different power level

regions. We have v0(λ?2) = λ?

2/(beaE1(a)) and H(v1(λ?2)) = λ?

2 with

H(x) = b1xea1+b1PSxE1 (a1 + b1PSx) (4.35)

and

a1 =l12σ2

SP1

, b1 =l12

P1. (4.36)

The optimal power value in the intermediate region is

p?(g22) =G−1 (λ?

2/(b1g22))− a1

b1g22(4.37)

withG(x) = exE1(x), ∀ x > 0. (4.38)

Proof. Regarding the reformulated optimization problem of (4.29) with objective func-tion (4.32).4 The Lagrangian of (4.29) is

L = Eg22

{ln

(1 +

P2(g22)g22

σ2S

)+ el12s1 E1(l12s1)

}− λ2

[Eg22 {P2(g22)} − PI l21

]. (4.39)

where the non-negative Lagrangian multiplier λ2 corresponds to the constraint (4.30).Herein, no Lagrange multipliers are assigned to the peak power constraints since theyare considered in the generalized KKT conditions.

According to the generalized KKT conditions for functional optimization [85, 87],the optimal solution P?

2 (g22) satisfies the following conditions:

dldP2(g22)

∣∣∣∣∣P2(g22)=P?

2 (g22)

≥ 0, P?

2 (g22) = PS

≤ 0, P?2 (g22) = 0

= 0, 0 < P?2 (g22) < PS

(4.40)

λ?2 ≥ 0 (4.41)

λ?2(Eg22 {P2(g22)} − PI l21

)= 0 (4.42)

Eg22 {P2(g22)} ≤ PI l21 (4.43)

4 The term el12s0 E1(l12s0) in (4.32) is independent of P2(g22). Therefore, we omit it in the algorithmicdesign since it does not affect the final result.



l = ln

(1 +

P2(g22)g22

σ2S

)+ el12s1 E1(l12s1)− λ?

2 (P2(g22)− PI l21) (4.44)

and the derivative of (4.40) w.r.t. P2(g22) is

dldP2(g22)

= b1g22ea1+b1g22P?2 (g22)E1(a1 + b1g22P?

2 (g22))− λ?2 (4.45)

where a1 and b1 are given in (4.36). Solving the KKT conditions, we obtain the optimalsolution P?

2 (g22). It is divided into three cases according to different values of g22:

1. P?2 (g22) = PS. According to the first inequality in (4.40), this holds under the

conditionb1g22ea1+b1PSg22 E1(a1 + b1PSg22) ≥ λ?

2 . (4.46)

2. P?2 (g22) = 0. According to the second inequality in (4.40), this holds under the

conditionb1ea1 E1(a1)g22 ≤ λ?

2 . (4.47)

3. 0 < P?2 (g22) < PS. According to the third equality in (4.40), we have the solution

P?2 (g22) = p?(g22) that satisfies the following nonlinear equation

b1g22ea1+b1g22 p?(g22)E1(a1 + b1g22p?(g22)) = λ?2 . (4.48)

As seen from (4.46), (4.47), and (4.48), P?2 (g22 = 0) = 0 holds if λ?

2 is positive. For thespecial case λ?

2 = 0, P?2 (g22 = 0) can be chosen as an arbitrary value in [0, PS] and

does not affect the objective function. Herein, we choose the solution P?2 (g22 = 0) = 0

for λ?2 = 0 to be consistent with the former case. In what follows, we only focus on

the discussion of the optimal power value P?2 (g22) for positive g22.

Using the definition of G(x) in (4.36), the equation (4.48) is reformulated into

b1g22G(a1 + b1g22p?(g22)) = λ?2 (4.49)

We prove in Appendix A.4 that G(x) is strictly monotonically decreasing in x forpositive and finite x. Consequently, the optimal solution of p?(g22) in (4.48) is givenin (4.37).

The solution of (4.37) is considered feasible if it is between zero and PS. Consider-ing that G(a1 + b1g22x) is also a monotonically decreasing function with x, x > 0,for positive a1, b1, and g22, we have the feasibility solutions (4.49) in the region0 < p?(g22) < PS as

b1g22G(a1 + b1g22PS) < λ?2 < b1g22G(a1). (4.50)

The effect in the variations of g22 on the optimal power P?2 (g22) is not explicitly

visible in (4.46), (4.47), and (4.50). In what follows, we aim at addressing this issue.


First, the left side of the first condition (4.46) is a nonlinear function in g22.We show that H(x) in (4.35) is a monotonically increasing function for x ≥ 0 andlimx→0 H(x) = 0 in Appendix A.5. Therefore, in order to satisfy (4.46), we requirethat g22 ≥ v1(λ

?2), where v1(λ

?2) is a lower bound on g22. The value of v1(λ

?2) is

equal to the root of H(x) = λ?2 and can be evaluated numerically. For the special case

λ?2 = 0, we have v1(λ

?2) = 0. Considering the second condition (4.47) for P?

2 (g22) = 0,we rewrite it as g22 ≤ v0(λ

?2) with v0(λ

?2) = λ?

2/(b1ea1 E1(a1)). This indicates that if thepower gain of the ST-SR link is smaller than v0(λ

?2), the optimal strategy is to switch

off the ST. Concerning the third case with 0 < P?2 (g22) < PS, where the power level

P?2 (g22) is given in (4.37), the condition (4.50) defines the complementary region given

by the first condition (4.46) and the second condition (4.47). Therefore, it correspondsto the case

v0(λ?2) < g22 < v1(λ

?2). (4.51)

For comparison, we apply the KKT conditions of functional optimization [87] tothe special case in [69, 71] that equivalent to P1 = 0. The same result is obtained.

Two questions arise from the above derived solution (4.34).

1. What is an efficient numerical method to obtain p?(g22) from (4.37)?

2. How to determine the parameter λ?2 in (4.34)?

In the following, we focus on answering these two problems.

The Algorithm for Solving (4.37) Eq. (4.37) is rewritten into

b1g22G(a1 + b1g22p?(g22)) = λ?2 (4.52)

According to (4.34), a feasible condition of p?(g22) ∈ (0, PS) is

v0(λ?2) < g22 < v1(λ

?2).

Moreover, G(a1 + b1g22x) is strictly monotonically decreasing in x for positive andfinite a1, b1 and g22. Combined with these two factors, the value of p?(g22) to solve(4.52) can be calculated by the bisection method. The bisection method repeatedlybisects an interval and selects a subinterval in which an optimum must lie for furtherprocessing until convergence. It has a linear rate of convergence [19].

Determination of λ?2 in (4.34) We discuss this issue under two circumstances. On the

one hand, if PS ≤ PI l21, the ST can always transmit with power PS while the averagepower constraint (4.30) is strictly satisfied, i.e., the optimal solution is constant andequal to PS and the computation of λ?

2 is not required. On the other hand, if PS > PI l21,using fg22(x) as the probability density function of g22, we have

Qavg = Eg22 {P?2 (g22)} =

∫ v1(λ?2)

v0(λ?2)

p?(x) fg22(x)dx + PS

∫ ∞

v1(λ?2)

fg22(x)dx. (4.53)


The optimal λ?2 should be chosen to satisfy Qavg = PI l21. We show that λ?

2 is finite andpositive in this case. The argument is proved by reduction to absurdity: If λ?

2 = 0, itfollows that v0(λ

?2) = v1(λ

?2) = 0. Integrating this into (4.34), we have P?

2 (g22) = PSfor any positive g22. Consequently, this yields Qavg = PS > PI l21 which violates(4.30). Similarly, we have P?

2 (g22) = 0 and Qavg = 0 if λ?2 is infinitely large, which

is obviously not optimal unless we consider the trivial scenario under the conditionPI > 0. Therefore, λ?

2 is finite and positive.In order to obtain λ?

2 to satisfy the equation (4.30) for the case PS > PI l21, wecalculate the derivative of Qavg over λ?

2 and obtain

dQavg

dλ?2

=

∫ v1(λ?2)

v0(λ?2)

dp?(x)dλ?

2fg22(x)dx

(a)< 0 (4.54)

where the inequality (a) uses the property that the optimal solution decreases as λ?2

increases. Given the strict monotonicity of Qavg over λ?2 in (4.54), the value of λ?

2 canbe calculated using the bisection method that has a linear rate of convergence [19].

Finally, we investigate the relation between the achievable performance and thesystem constraints by using the optimal solution (4.34). We show that the achievablerate remains constant provided either PS or PI is fixed and increasing PI or PS beyondsome limit given by PI,2 and PS,2, respectively.

On the one hand, if PI keeps increasing and satisfies PI ≥ PS/l21, the optimalpower is equal to PS which results in a constant achievable rate R in this case. There-fore, we have PI = PS/l21. On the other hand, we have

b1g22ea1+b1PSg22 E1(a1 + b1PSg22) < b1g22/(a1 + b1PSg22) < 1/PS (4.55)

by applying (A.15) in Appendix A.4 to the left side of (4.46). This means that given acertain λ?

2 , increasing PS beyond some boundary point PS,2, i.e., the constraint (4.46) isalways violated. The optimal power never reaches the peak power limit PS. Therefore,increasing PS does not affect the achievable rate. The threshold PS,2 can be approxi-mated by PS that is obtained from the following equality:∫ ∞

v0=1/(b1ea1 G(a1)PS)p?(x) fg22(x)dx = PI l21. (4.56)

4.2.2.2 Suboptimal Power Allocation I

The optimal power allocation strategy in (4.34) requires two numerical computationsteps: the computation of p?(g22) and the calculation of λ?

2 . The value of p?(g22)needs to be calculated for every time instant, while the value of λ?

2 needs to be up-dated with changing statistical channel parameters. Hence, calculating the optimalsolution is time-consuming. Moreover, due to the lack of the optimal solution inclosed form, deriving the analytical performance is intractable. In order to reducethe algorithm complexity, we propose two suboptimal low complexity strategies inSection 4.2.2.2 and Section 4.2.2.3. The first one, named double threshold waterfilling


(DT-WF), is based on an approximation of the optimal solution. The second strategy,named double threshold constant-power waterfilling (DTCP-WF), further simplifiesDT-WF. Additionally, benefited from the analytical expression of the solutions, theperformance of two suboptimal solutions is given in closed form.

The real-time computation of p?(g22) requires numerically solving an equation inthe form of

G(x) = C

where C is some positive constant. In order to obtain a low-complexity strategy, weresort to an approximate solution of x. Particularly, we use both the upper-bound andthe lower-bound of G(x) in (A.15) as two approximations

G(x) ≈ 1x

(4.57)

G(x) ≈ 1x + 1

(4.58)

respectively. By checking the approximation accuracy of both expressions, we observethat both approximations are close to G(x) when x is large. Among the two of them,the approximation G(x) ≈ 1/(x + 1) is close to G(x) in general. Therefore, we applyG(x) ≈ 1/(x + 1) in the remainder of this section. Specifically, p?(g22) in (4.37) isapproximated by

p?(g22) = v?2 −a1 + 1b1g22

(4.59)

where the calculation of v?2 will be addressed later.Similarly, we use G(x) ≈ 1/(x + 1) to approximate the conditions in (4.34) un-

der which P?2 (g22) is equal to 0, p?(g22), PS, respectively. After some mathematical

manipulations, the suboptimal power control strategy DT-WF is obtained as

v?2 > PS : P?2 (g22) =

0, g22 ≤

a1 + 1b1v?2

p?(g22),a1 + 1b1v?2

< g22 <a1 + 1

b1(v?2 − PS

)PS, otherwise

(4.60)

0 < v?2 ≤ PS : P?2 (g22) =

0, g22 ≤a1 + 1b1v?2

p?(g22), otherwise.(4.61)

One crucial question in calculating P?2 (g22) is how to determine v?2 in (4.60) and

(4.61). If PS ≤ PI l21, the suboptimal power is P?2 (g22) = PS. Hence, the computation

of v?2 is not required. Otherwise in the case PS > PI l21, v?2 is finite and positive. In thefollowing, we propose an efficient method to calculate v?2 .

Proposition 4.2.3. For PS > PI l21, the positive parameter v?2 is uniquely chosen to satisfy

F2

(v(?)2

)= PI l21 (4.62)


with F2(x) given byF2(x) = f2(x)− f2(x− PS) (4.63)

and the definition of f2(x) as

f2(x) =

xe−λbx − λbE1

(λbx

), x > 0

0, otherwise.(4.64)

with λb = (a1 + 1)l22/b1. The value of v?2 can be obtained by applying the Newton’s method

v(n+1)2 = v(n)2 −

F2

(v(n)2

)− PI l21

F′2(v(n)2 )

(4.65)

after convergence. The sequence (4.65) satisfies the standard assumptions of the local conver-gence theorem for the Newton’s method [67].

The initial point of the iteration is provided by

v(0)2 = − (a1 + 1)l22

b1 ln (PI l21/PS). (4.66)


A theoretical performance analysis is desirable in designing the algorithm sincethe performance can be evaluated without running numerous simulations. Consider-ing the suboptimal solution (4.60) and (4.61) in which the explicit connection betweenP?

2 (g22) and the instantaneous CSI g22 is shown, deriving an analytical performanceresult is feasible. The main result is summarized as follows.

Proposition 4.2.4. Given the functions r1 (λ, p) and r2 (λ) in (A.31) and (A.39) and

g1 =(a1 + 1)P1

(v?2 − PS)l12, v?2 > PS (4.67)

g0 =(a1 + 1)P1

v?2 l12, v?2 > 0 (4.68)

the achievable rate of the suboptimal strategy DT-WF is

RDT-WF =

{r1 (g1, PS)− r2 (g1) + r2 (g0) , v?2 > PS

r2 (g0) , 0 < v?2 ≤ PS.(4.69)


4.2.2.3 Suboptimal Power Allocation II

In DT-WF, the power level p?(g22) is calculated in real-time, since it depends on the in-stantaneous CSI g22 in (4.59). In order to further reduce the computational complexity,


we exploit non adaptive power transmission in the region where p?(g22) is allocated.The idea is motivated by the conventional constant-power waterfilling strategy [138]where the transmitter allocates constant power to the channels with nonzero powerby optimal waterfilling and zero power in the remaining channels. However, differentfrom the conventional idea, DTCP-WF is designed based on the thresholds of DT-WFinstead of the optimal solution since the computation of the thresholds in the optimalsolution is more time-consuming.

In DTCP-WF, zero and the peak power value PS are allocated in the same regionsas in DT-WF, otherwise constant power is allocated. Consequently, the form of DTCP-WF is the same as (4.60) and (4.61) except that p?(g22) is replaced by the constantpower level p?c .

The calculation of p?c is performed as follows. For the case PS ≤ PI l21, the optimalsolution is reduced to transmitting with power PS in every time instant, thus the cal-culation of p?c is not required. For the case PS > PI l21, the constant power level p?c (v?2)is chosen to satisfy the average power constraint (4.30). After some mathematicalmanipulations, we obtain

v?2 > PS, P?2 (g22) =

0, b1g22 ≤

a1 + 1v?2

PS, b1g22 ≥a1 + 1

v?2 − PS

p?c (v?2), otherwise

. (4.70)

0 < v?2 ≤ PS, P?2 (g22) =

0, b1g22 ≤a1 + 1

v?2p?c (v?2), otherwise

. (4.71)

where

p?c =

PI l21 − PSe

− (a1+1)l22b1(v?2−PS)

e− (a1+1)l22

b1v?2 − e− (a1+1)l22

b1(v?2−PS)

, v?2 > PS

PI l21

e− (a1+1)l22

b1v?2

, 0 < v?2 ≤ PS.

(4.72)

The performance of DTCP-WF can be derived similarly as for DT-WF using theexpression r1 (λ, p) given in (A.31). For completeness, we directly provide the result:

RDTCP-WF =

{r1 (g1, PS)− r1 (g1, p?c ) + r1 (g0, p?c ) , v?2 > PS

r1 (g0, p?c ) , 0 < v?2 ≤ PS(4.73)

where g1 and g0 are given in (4.67) and (4.68), respectively.


4.2.2.4 Multiple Primary Links Scenario

The difficulty in solving the considered optimization problem (4.29) arises from thepartial CSI related to the primary links. The existence of multiple primary links inthe spectrum-sharing system brings more challenges to the optimization problem. Inthis section, we discuss the extension to the system containing multiple primary linksand show that the optimal power allocation strategy can be similarly derived for thegeneral case.

Specifically, we assume there are M PT-PR links coexisting with one ST-SR linkin the system. The mth PT transmits at the power level P1,m, ∀m = 1, . . . , M. TheCSI from the mth PT to the SR is denoted by g12,m with its mean 1/l12,m. Similarly,the CSI from the ST to the mth PR link is g21,m with its mean 1/l21,m. The primarymessages are independent. Consistent with the CSI assumption in the single primarylink scenario, the ST is assumed to know the instantaneous CSI of g22 and statisticalCSI l12,m and l21,m, ∀m = 1, . . . , M. All channels are assumed to be stationary, ergodic,and mutually independent.

We first define a new variable g2 denoting the sum of the interference power fromM PTs at the SR as

g2 =M∑

m=1

P1,mg12,m (4.74)

where P1,mg12,m, ∀m = 1, . . . , M, is independent exponential random variable withexpected value given as

1lm

=P1,m

l12,m. (4.75)

In a practical system, the values of lm, ∀m = 1, . . . , M, concerning M primary linksare distinct. Therefore, for distinct lm, ∀m = 1, . . . , M, the PDF of g2 is [72, Eq. (3)]

fg2(g2) =M∑

m=1

Am lme−lmg2 (4.76)

with

Am =M∏

u=1,u 6=m

11− lm/lu

. (4.77)

Using the definition of g2 in (4.74), the optimization problem is reformulated as

(P2) maxP2(g22)

Eg2,g22

{ln

(1 +

P2(g22)g22

g2 + σ2S

)}(4.78)

s.t. P2(g22) ≤ PS, ∀g22 ≥ 0P2(g22) ≥ 0, ∀g22 ≥ 0Eg21,m,g22 {P2(g22)g21,m} ≤ PI,m

∀m = 1, . . . , M


where PI,m is the interference power limit of the mth PR. We can show that (4.78) is aconvex optimization problem.

Using the independence between the CSI of different links, the last constraints in(4.78) can be simplified to a single constraint

Eg22 {P2(g22)} ≤ PI,MU (4.79)

where PI,MU = PI,ml21,m holds and the index m is selected according to

m = arg minm∈{1,...,M}

{PI,ml21,m} (4.80)

In order to derive the optimal power allocation strategy, we need to reformulatethe objective function in (4.78) as an explicit function of the available CSI. Specifically,inserting (4.76) into the objective function of (4.78)), we have

Eg1, g22

{ln

(1 +

P2(g22)g22

σ2S + g2

)}

= Eg22

{ M∑m=1

Am

(ln

(1 +

P2(g22)g22

σ2S

)+ elm(σ2

S+P2(g22)g22)E1

(lm(

σ2S + P2(g22)g22

))− elmσ2

S E1

(lmσ2

S

))}. (4.81)

We integrate (4.79) and (4.81) into (4.78). Applying functional optimization theory[85, 87], and after some mathematical manipulations, we obtain the optimal solution

P?2 (g22) =

0, H1(0) ≤ λ?

2,MU

p?MU(g22), H1(PS) < λ?2,MU < H1(0)

PS, otherwise

(4.82)

where the function H1(x) is defined as

H1(x) =M∑

m=1

Am lmg22elm(σ2S+xg22)E1

(lm(

σ2S + xg22

))(4.83)

and p?MU(g22) is the root of the equation

H1 (x) = λ?2,MU (4.84)

with λ?2,MU as the optimal value of the non-negative Lagrangian multiplier that corre-

sponds to the constraint (4.79).


In order to solve p?MU(g22) from (4.84) efficiently, we check the derivative of H1 (x)w.r.t. x, x ∈ (0, PS), within its feasibility region given in (4.82). Specifically, we have

H′1(x) =

∂2Eg2

{ln(1 + xg22/(σ2

S + g2))}

∂x2 = −Eg2

{1(

(σ2S + g2)/g22 + x

)2

}< 0

(4.85)

for any positive x. Hence, H1(x) is a strictly monotonically decreasing function inx in the feasible region. Thus, p?MU(g22) can be obtained from (4.84) using simplenumerical methods, e.g., the bisection method.

4.2.2.5 Performance Comparison

We assess the achievable rate in bits/s/Hz. The channel power gain of each fadinglink gij, i, j = 1, 2 follows an exponential distribution with l12 = 2, l21 = 3, and l22 = 1.The noise variance of the SU is σ2

S = 1. All power values in dB are given relative tothe reference level of value 1. The transmit power of the PT is 10dB.

The performance of five power allocation strategies is compared.

• Optimal algorithm: The power level value is mapped from g22 as given in (4.34).

• DT-WF: The power level value is a function of g22 as given in (4.60) and (4.61).The analytical performance is provided by (4.69).

• DTCP-WF: The power level is matched to g22 as given in (4.70 ) and (4.71), wherep?(g22, v?2) from DT-WF is replaced by p?c in (4.72). The analytical performanceis provided by (4.73).

• Reference algorithm 1: The first reference algorithm was developed in [69, 71]. Itcan be considered as the result of the special case of (4.29) with P1 = 0, i.e., theinterference from the PT is ignored.

• Reference algorithm 2: The second reference algorithm is a non-adaptive strategyaiming at satisfying the system constraints at every time instant:

P2(g22) =

{PS, PS ≤ PI l21

PI l21, otherwise.(4.86)

Figure 4.4 plots the achievable rate R versus the peak transmit power constraintPS. The left hand side subfigure shows the performance under different interferencepower constraints PI . The detailed explanations of the legend are given below thefigure. We have several observations. Firstly, concerning the achievable performance,the analytical results match the numerical results for the suboptimal solutions. Sec-ondly, the near-optimality of the suboptimal solutions is validated. The gain overthe reference non-adaptive power transmission is large for each case of PI starting


0 5 10 15 20 250

0.5

1

1.5

2

Transmit power constraint limit PS [dB]

Ach

ieva

ble

rate

R[b

its/

s/H

z] PI=10 dBPI= 5 dBPI= 0 dB

10 15 20 25 300.7

0.8

0.9

1

1.1

Transmit power constraint limit PS [dB]

Ach

ieva

ble

rate

R[b

its/

s/H

z]

Opt. alg.DT-WFDTCP-WFRef. alg. 1Ref. alg. 2.

Figure 4.4: Achievable rate of the SU vs. peak transmit power constraint limit PSwith P1 = 15 dB. The lines/markers of the same color share the samesetup parameter PI . Solid lines: optimal solution; dashed lines: DT-WF (sim.); circles: DT-WF (analytical); dotted lines: DTCP-WF (sim.);squares: DTCP-WF (analytical); diamond: ref. alg. 1; dash-dotted lines:ref. alg. 2. In the right hand side subfigure, PI = 5 dB holds.

−10 −5 0 5 10 15 200

0.5

1

1.5

2

Interference power limit PI [dB]

Ach

ieva

ble

rate

R[b

its/

s/H

z] PS=15 dBPS= 5 dB

0 2 4 6 8 10 12 14

0.6

0.8

1

1.2

1.4

1.6

Interference power limit PI [dB]

Ach

ieva

ble

rate

R[b

its/

s/H

z]

Opt. alg.DT-WFDTCP-WFRef. alg. 1Ref. alg. 2

Figure 4.5: Achievable rate of the SU vs. peak transmit power constraint limit PIwith P1 = 15 dB. The lines/markers of the same color share the samesetup parameter PS. Solid lines: optimal solution; dashed lines: DT-WF (sim.); circles: DT-WF (analytical); dotted lines: DTCP-WF (sim.);squares: DTCP-WF (analytical); diamonds: ref. alg. 1; dash-dotted lines:ref. alg. 2. In the right hand side subfigure, PS = 15 dB holds.

at certain PS. Finally, R increases with an increase of PS up to some boundary, be-yond which it remains constant. The boundary can be calculated with (4.56). Thedetailed comparison on the performance are shown in the right hand side subfigure


10 15 20 25 300

0.2

0.4

0.6

0.8

1

Primary transmit power P1 [dB]

Ach

ieva

ble

rate

R[b

its/

s/H

z]

PI = 0 dB and PS = 10 dB

Opt. alg.DT-WF (sim.)DT-WF (analytical)DTCP-WF (sim.)DTCP-WF (analytical)Ref. alg. 1Ref. alg. 2.

10 15 20 25 300

0.5

1

1.5

2

Primary transmit power P1 [dB]

Ach

ieva

ble

rate

R[b

its/

s/H

z]

PI = 5 dB and PS = 15 dB

Figure 4.6: Achievable rate of the SU vs. primary transmit power P1.

for the case of PI = 5 dB. The proposed optimal and suboptimal strategies outperformthe reference ones especially for large values of PS. Considering that our algorithmsrequire no additional instantaneous CSI compared to the reference algorithms, thegain is mainly due to taking the interference term P1 into account. Besides, we notethat the performance loss of the suboptimal strategies compared to the optimal onesincreases slightly from 12 dB to 14 dB. The reason is that the approximation (4.59) in-curs larger approximation errors for large values of PS, consequently yielding a largerperformance gap between the proposed strategies and the optimal one.

Figure 4.5 depicts the achievable secondary rate R versus the interference powerconstraint PI . In the left hand side, the performance is shown under different trans-mit power constraints PS. The simulation and analytical results for the suboptimalsolutions match. The near-optimality of the suboptimal solutions is also verified.In addition, it is shown that the achievable rate saturates if PI increases beyond theboundary PS/l21 provided that PS is fixed. The detailed performance comparison isdepicted in the right hand side subfigure for the case of PS = 15 dB. It is verifiedthat the proposed optimal and suboptimal strategies outperform the reference onesespecially for small values of PI .

Figure 4.6 examines the effect of the primary transmit power P1 on the achievablesecondary rate R. We set in left hand side subfigure PS = 10 dB and PI = 0 dB. Wecompare the performance of all proposed strategies and the reference ones. The near-optimality of the proposed strategies is validated over a large region of P1. Comparedto our strategies, the reference algorithm 1 suffers from a performance degradationin the high P1 region due to its ignorance of P1. We remark that the proposed algo-rithms require no additional instantaneous CSI compared to the reference algorithm1. Similar observations can be made in the right hand side subfigure with PS = 15 dBand PI = 5 dB are plotted. Furthermore, both subfigures validate the accuracy of theanalytical results for the suboptimal strategies.


4.3 Power Control with Outage Probability Constraint

In this subsection, we investigate the gain achieved by the SUs to exploit the partialCSI related to the primary transmission links. Specifically, revising the system modelgiven in Figure 4.1, we assume the ST and the SR have the knowledge of instantaneousCSI g22 and g12 and statistical CSI l11 and l21. Two constraints are imposed on thesystem design. Firstly, we limit the probability that the instantaneous rate of theprimary transmission is smaller than γp (bits/s/Hz) below a threshold εp ∈ [0, 1]

Pr

{log2

(1 +

P1g11

P2(g12, g22)g21 + σ2P

)< γp

}≤ εp (4.87)

Secondly, the power of the ST is non-negative and limited by PS:

0 ≤ P2(g12, g22) ≤ PS, ∀ g12 ≥ 0, g22 ≥ 0. (4.88)

Consequently, the optimization problem aiming at maximizing the average achiev-able rate of the secondary link subject to the aforementioned constraints is

maxP2(g12,g22)

Eg12, g22

{ln

(1 +

P2(g12, g22)g22

P1g12 + σ2S

)}(4.89)

s.t. Pr

{log2

(1 +

P1g11

P2(g12, g22)g21 + σ2P

)< γp

}≤ εp

0 ≤ P2(g12, g22) ≤ PS, ∀ g12 ≥ 0, g22 ≥ 0.

As shown in (4.89), the objective function is a maximization of a concave function ofP2(g12, g22) and the constraint (4.88) is an affine function of P2(g12, g22). Thus, theconvexity of (4.89) depends on the convexity of the constraint (4.87) which is unclearin its current form. Therefore, we reformulate the outage probability as

Pr

{log2

(1 +

P1g11

P2(g12, g22)g21 + σ2P

)< γp

}(a)= Pr

{g11 <

γthσ2P

P1+

γthP2(g12, g22)

P1g21

}

=Eg12,g22

{Pr

{g11 <

γthσ2P

P1+

γthP2(g12, g22)

P1g21

∣∣∣∣∣P2(g12, g22)

}}

(b)= Eg12,g22

1− e−l11γthσ2

PP1

1

1 + l11γthl21P1

P2(g12, g22)

(c)= 1− e−

l11γthσ2P

P1 Eg12,g22

{1

1 + aP2(g12, g22)

}(4.90)

4.3. Power Control with Outage Probability Constraint 67

where, in (a), we use γth = 2γp − 1. In (b), the result in [63, Appendix I] is applied. In(c), the deterministic terms are moved out of the expectation operator and the positivea = l11γth/(l21P1) is used for notational simplification.

Inserting (4.90) into (4.87), the outage probability constraint is converted into

Eg12,g22

{1

1 + aP2(g12, g22)

}≥ ε (4.91)

where the parameter ε is

ε = el11γthσ2

PP1 (1− εp), ε ∈ [0, 1]. (4.92)

The constraint (4.91) is restricting a convex functional on P2(g12, g22) to a value largerthan a certain threshold, which is non-convex. Therefore, (4.89) is a non-convex opti-mization problem. Using the variable t as (4.7), we represent P2(g12, g22) as P2(t) andreformulate the optimization problem (4.89) as

maxP2(t)

R (P2(t)) (4.93)

s.t. 0 ≤ P2(t) ≤ PS, ∀ t > 0Qc (P2(t)) ≥ ε

where R (·) is given in (4.9) and Qc (·) is defined as

Qc (x(t)) = Et

{1

1 + ax(t)

}(4.94)

Due to the equivalence between (4.89) and (4.93), we focus on finding the optimalsolution to the non-convex problem (4.93) for the remainder of the subsection.

4.3.1 Optimal Power Allocation

In general, a non-convex optimization problem is difficult to solve. In this section,we aim at designing the optimal solution to the non-convex problem (4.93). To thisend, we first show that strong duality holds in spite of non-convexity and thus theKKT conditions are necessary optimality conditions. Using this property, the KKTsolutions are derived. The optimality of each KKT solution is then characterized.

Concerning problem (4.93) and ε ∈ [0, 1], it is easy to verify that the solution forε = 1 is P?

2 (t) = 0. Therefore, the remaining task is to solve (4.93) for 0 ≤ ε < 1.In Appendix A.8, we apply the theorems in [127, 138] and show that strong dualityholds between (4.93) and its dual problem. This motivates us to solve the problem inthe Lagrangian dual domain.

Since the optimization problem (4.93) has differentiable objective and constraintfunctions for which strong duality holds, the optimal solution can be attained bysolving the KKT conditions [111]. Assuming the non-negative parameter v? is the


inverse of an optimal Lagrangian multiplier λ? and applying the KKT conditions forfunctional optimization [85, 87] to (4.93), we derive the KKT solutions as follows.

Proposition 4.3.1. Solving the KKT conditions of (4.93) yields

P?2 (t) = PS, if t ≤ T1 (v?) (4.95)

P?2 (t) = 0, if t ≥ T2 (v?) (4.96)

P?2 (t) = x1 (t, v?) ∨ P?

2 (t) = x2 (t, v?) , if t ∈ Ft (4.97)

where

x1 (t, v) =1− 2v +

√1− 4v + 4avt

2av(4.98)

x2 (t, v) =1− 2v−

√1− 4v + 4avt

2av(4.99)

T1 (v) =(1 + aPS)

2

av− PS (4.100)

T2 (v) =va

. (4.101)

The feasible regions Ft to constrain P?2 (t) = x1(t, v?) ∈ (0, PS) and P?

2 (t) = x2(t, v?) ∈(0, PS) are analyzed in Appendix A.10. If 0 ≤ ε ≤ 1/(1 + aPS), the solution is P?

2 (t) = PS,cf. (4.91). Thus the calculation of v? is not required. If 1/(1 + aPS) < ε < 1, v? needs to beselected to let the constraint Qc (P?

2 (t)) ≥ ε be satisfied with equality.


Therefore, we only need to focus on deriving the non-trivial optimal solution for1/(1 + aPS) < ε < 1. Note that KKT conditions are necessary but not sufficient op-timality conditions under such circumstances, i.e., the primal and dual optimum areamong the KKT solutions. We thus characterize the optimality of the KKT solutionsby checking sufficient conditions [14, 58].

Lemma 4.3.1. Given a non-negative v?, P?2 (t) = x1 (t, v?) yields a local minimum and

P?2 (t) = x2 (t, v?) yields a local maximum in the corresponding feasible region of t given in

Appendix A.10.


Proposition 4.3.1 combined with Lemma 4.3.1 implies that the optimal solution to(4.93) is made of three functions: two functions at the boundary points, i.e., P?

2 (t) = PSand P?

2 (t) = 0, and the function at the interior points P?2 (t) = x2 (t, v?). The regions

of t for these functions are given in (4.95), (4.96), and Appendix A.10, respectively.We assume that the optimal solution is a piecewise continuous function which is acommon assumption in optimal control theory, e.g., [74]. In addition, there is no re-movable discontinuity since this kind of discontinuity can be removed to make thefunction continuous at the discontinuity. Therefore, the finite number of discontinu-ities are located in the corresponding feasible region of t for each function. Figure 4.7


0 2 4 6 8 10

0

5

10

15 P2(t) = PS

P2(t) = 0

P2(t) = x2(t, v)

T1(v) T2(v)

t

Case 1− 2v? ≥ 2av?PS

0 2 4 6 8 10

0

5

10

15

20

P2(t) = PS

P2(t) = 0

T2(v)T1(v)

t

Case 1− 2v? ≤ 0

10.5 11 11.5 12 12.5

0

2

4

6

8

10

P2(t) = PS

P2(t) = 0P2(t) = x2(t, v)

T3(v)T2(v) T1(v)

t

Case 0 < 1− 2v? ≤ av?PS

0 2 4 6 8 10

0

10

20P2(t) = PS

P2(t) = 0

P2(t) = x2(t, v)

T3(v) T1(v) T2(v)

t

Case av?PS < 1− 2v? < 2av?PS

Figure 4.7: Exemplified candidate solutions for different cases of v?. The functionsT1(v), T2(v), and T3(v) are given in (4.100), (4.101), and (A.51), respec-tively.

exemplifies the candidate solutions for the four cases of v?. The feasible regions of tfor the three functions are not mutually exclusive under the following circumstances:

• The regions of t for P?2 (t) = PS and P?

2 (t) = 0 overlap if 1− 2v? ≤ av?PS.

• The regions of t for P?2 (t) = PS and P?

2 (t) = x2 (t, v?) overlap if 0 < 1− 2v? <2av?PS.

Due to the fact that the discontinuities can be arbitrarily located in the correspondingfeasible region of t for each function and t is continuous, there is an infinite number ofcandidate solutions. In order to further reduce the set of candidate optimal solutions,we provide an additional characteristic of the optimal solution to (4.93).

Lemma 4.3.2. The optimal solution P?2 (t) to (4.93) is monotonically decreasing in t.



Lemma A.12 indicates that the optimal power allocation strategy is to allocate alarge amount of power when the CSI of the ST-SR link to interference plus noise ratiois high and allocate a small amount of power when this ratio is low. This propertyis reasonable since the ST needs to allocate more power during favorable channelstates in order to maximize the achievable rate of the secondary link while keepingthe outage probability of the PU under a certain limit.

Remark: Lemma 4.3.1 can be partly verified by Lemma 4.3.2. More particularly,a solution containing P?

2 (t) = x1 (t, v?) is not the optimal solution since x1 (t, v?) isan increasing function of t. However, Lemma 4.3.1 additionally shows that a solutioncontaining P?

2 (t) = x2 (t, v?) might be a optimal solution. This part cannot be vali-dated by Lemma 4.3.2 since it only provides a necessary optimality condition tailoredto the problem (4.93).

Given the above-mentioned characteristics of the optimal solution, we present theoptimal solution of (4.93) in the following theorem:

Theorem 4.3.1. The optimal solution of (4.93) is given as1) P?

2 (t) = PS, if 0 ≤ ε ≤ 1/(1 + aPS).2) P?

2 (t) = 0, if ε = 1.3) If 1/(1 + aPS) < ε < 1, the optimal solution P?

2 (t) is divided into three cases:

• If 1− 2v? ≥ 2av?PS,

P?2 (t) =

PS, if 0 ≤ t ≤ T1 (v?)x2 (t, v?) , if T1 (v?) < t < T2 (v?)0, otherwise.

(4.102)

The value of v? is chosen to let the equality Qc (P?2 (t)) = ε hold.

• If 1− 2v? ≤ 0,

P?2 (t) =

{PS, if 0 ≤ t ≤ T2 (v?)0, otherwise.

(4.103)

The value of v? is chosen to let the equality Qc (P?2 (t)) = ε hold.

• If 0 < 1− 2v? < 2av?PS, the solution P?2 (t) is a function of v? where

v? = arg maxv{R (P2(t))} (4.104)

with

P2(t) =

PS, if 0 ≤ t ≤ γ(v)x2(t, v), if γ(v) < t < T2(v)0, otherwise.

(4.105)


where the non-negative γ(v) is in [T3(v), T1(v)] and it is chosen to satisfy Qc (P?2 (t)) =

ε. The functions T1(v), T2(v), and T3(v) are given in (4.100), (4.101), and (A.51),respectively.

Proof. It is easy to verify that the optimal solution is P?2 (t) = PS and P?

2 (t) = 0 for0 ≤ ε ≤ 1/(1 + aPS) and ε = 1, respectively. Therefore, we only focus on the case1/(1 + aPS) < ε < 1. In the following, the optimal solution is provided as a functionof v?. Incorporating them into the outage probability constraint (4.91) yields the leftside as the function of v? only. Then v? can be obtained by solving (4.91) with equality.

1. 1− 2v? ≥ 2av?PS:

The case is exemplified in the upper-left subfigure in Figure 4.7. We can ver-ify that T1(v?) < T2(v?) holds. The solutions P?

2 (t) = 0, P?2 (t) = PS, and

P?2 (t) = x2(t, v) have non-overlapping feasible regions of t. Therefore, the op-

timal solution is given in (4.102) as the unique piecewise continuous functionconsisting of P?

2 (t) = 0, P?2 (t) = PS, and P?

2 (t) = x2(t, v).

2. 1− 2v? ≤ 0:

One example under this case is given in the upper-right subplot in Figure 4.7.For this case T2(v?) < T1(v?) holds. The optimal solution is a piecewise contin-uous function combined with P?

2 (t) = PS and P?2 (t) = 0 with the feasible region

of t as t ≤ T1 (v?) and t ≥ T2 (v?), respectively. The discontinuity can thus bechosen in [T2 (v?) , T1 (v?)]. However, if the optimal power allocation strategy issuch a binary power switching scheme, there exists a unique discontinuity to letthe outage probability constraint be satisfied with equality, i.e., the value of thediscontinuity is fixed. Without loss of generality, we give the optimal solutionin (4.103) where the discontinuity is located at T2 (v?).

3. 0 < 1− 2v? < 2av?PS:

Two different examples are given in the lower-left and lower-right subplots inFigure 4.7 in which T2(v?) ≤ T1(v?) and T1(v?) < T2(v?) holds, respectively.The feasible regions of t for P?

2 (t) = PS, P?2 (t) = 0, and P?

2 (t) = x2(t, v) aret ≤ T1 (v?), t ≥ T2 (v?), and T3 (v?) < t < T2 (v?), respectively, and they overlap.Therefore, the optimal solution can be a piecewise continuous monotonicallydecreasing function composed of the three functions in which the discontinu-ities are located at arbitrary points in the overlapping region. The influence onthe objective function of the variation of the discontinuities is difficult to eval-uate, resulting in difficulty in further reducing the search space of the optimalsolution. We give the solution as a function of v and claim that the optimal solu-tion can be obtained by solving an optimization problem over a single boundedvariable v.

• T1 (v) ≥ T2 (v): The lower-left subplot in Figure 4.7 indicates one example.

a) For a discontinuity at γ(v) when switching from P?2 (t) = PS to P?

2 (t) =x2(t, v) for an increase of t, it is located at an arbitrary point in theregion [T3 (v) , T2 (v)]. Note that there is no discontinuity switching


from P?2 (t) = x2(t, v) to P?

2 (t) = PS for an increase of t since it violatesthe optimality condition in Lemma 4.3.2. In addition, the candidatesolution switches from P?

2 (t) = x2(t, v) to P?2 (t) = 0 at T2(v). Due

to Lemma 4.3.2, only P?2 (t) = 0 holds for t ≥ T2 (v). The solution is

expressed in (4.105) with γ(v) ∈ [T3 (v) , T2 (v)] and γ(v) is chosen tosatisfy the constraint Qc (P2(t)) = ε.

b) For a discontinuity at γ(v) when switching from P?2 (t) = PS to P?

2 (t) =0 for an increase of t, then γ(v) is located at an arbitrary point in theregion [T2 (v) , T1 (v)]. Similar to the last case, only P?

2 (t) = 0 holds fort ≥ γ (v) due to the optimality condition in Lemma 4.3.2. The solutionis also expressed in the form of (4.105) with γ(v) ∈ [T2 (v) , T1 (v)] andγ(v) is chosen to let the constraint Qc (P2(t)) = ε be satisfied.

Combined with the above two cases, the power allocation strategy is givenin (4.105) with γ(v) ∈ [T3 (v) , T1 (v)]. The value γ(v) should be chosen tosatisfy Qc (P2(t)) = ε.

• T1 (v?) < T2 (v?): As shown in the exemplified in the lower-right subplotin Figure 4.7, a discontinuity at γ(v) when switching from P?

2 (t) = PS toP?

2 (t) = x2(t, v) for an increase of t can be located at an arbitrary pointin the region [T3 (v) , T1 (v)]. According to Lemma 4.3.2, there exists nodiscontinuity when switching from P?

2 (t) = x2(t, v) to P?2 (t) = PS. Further-

more, the candidate solution switches from P?2 (t) = x2(t, v) to P?

2 (t) = 0 atT2(v). For t ≥ T2 (v), the solution is P?

2 (t) = 0. The solution also matchesthe form in (4.105). The value γ(v) ∈ [T3 (v) , T1 (v)] is chosen to satisfy theconstraint Qc (P2(t)) = ε.

The proof is concluded by combining the results discussed in each case.

For the third case 1/(1+ aPS) < ε < 1 in Theorem 4.3.1, the value of v? is critical todetermine the optimal solution. For example, P?

2 (t) in (4.102) and (4.103) is a functionof v? since T1 (v?), T2 (v?), T3 (v?), and x2(t, v?) are all functions of v?, incorporatingthem into Qc (P?

2 (t)) results in Qc (·) being dependent on v?. We can then obtain v?

by solving Qc (P?2 (t)) = ε. Similarly, P2(t) in (4.105) is also provided as a function of v.

Taking it into Qc (P2(t)) and solving Qc (P2(t)) = ε yields a feasible v. Incorporatingv into (4.105), we obtain a feasible solution P2(t). Finally, searching over all feasiblev, we select v? corresponding to P?

2 (t) that results in the largest secondary achievablerate in (4.104). Briefly speaking, (4.104) is a nonlinear maximization problem over asingle bounded variable v for 0 < 1− 2v < 2avPS. Some optimization toolboxes can beapplied for its efficient implementation, e.g., fminbnd in Matlab©.

4.3.2 Suboptimal Power Allocation

The optimal strategy requires a complex computation for the determination of v?, es-pecially for the case 0 < 1− 2v? < 2av?PS since the optimization (4.104) is required.Furthermore, the achievable performance is hard to evaluate analytically. This neces-sitates the development of low complex strategies with performance analysis.


4.3.2.1 Suboptimal Power Allocation Strategy I

The suboptimal strategy I results from simplifications of the optimal strategy P?2 (t).

The simplifications are twofold: Firstly, the nonlinear part x2(t, v) in P?2 (t) is replaced

with its first-order Taylor polynomial x2(t, v); secondly, the time-consuming optimiza-tion in (4.104) is replaced by a selection of v among two extreme cases. The resultingsolution P?

2 (t) is given as follows.

Proposition 4.3.2. Given a certain non-negative v?, the suboptimal solution I of (4.93) is

1. P?2 (t) = PS, if 0 ≤ ε ≤ 1/(1 + aPS).

2. P?2 (t) = 0, if ε = 1.

3. If 1/(1 + aPS) < ε < 1, we distinguish between three cases:

• If 1− 2v? ≥ 2av?PS,

P?2 (t) =

PS, if 0 ≤ t ≤ T1 (v?)x2 (t, v?) , if T1 (v?) < t < T2 (v?)0, otherwise.

(4.106)

• If 1− 2v? ≤ 0,

P?2 (t) =

{PS, if 0 ≤ t ≤ T2 (v?)0, otherwise.

(4.107)

• If 0 < 1 − 2v? < 2av?PS, the solution P?2 (t) is chosen among two candidate

solutions P2,1(t) and P2,2(t), that results in the larger objective function

P?2 (t) = arg max

P2(t)

{R(P2(t)) : P2(t) ∈

{P2,1(t), P2,2(t)

}}(4.108)

The functions P2,1(t) and P2,2(t) are functions of v1 and v2, respectively

P2,1(t) =

PS, if 0 ≤ t ≤ T1 (v1)

x2 (t, v1) , if T1 (v1) < t < T2 (v1)

0, otherwise.(4.109)

P2,2(t) =

PS, if 0 ≤ t ≤ T3 (v2)

x2 (t, v2) , if T3 (v2) < t < T2 (v2)

0, otherwise.(4.110)

In (4.106)-(4.110), we use

x2(t, v) =1

1− 2v

(va− t)

(4.111)


which is a linear function of t and the functions T1(v), T2(v), and T3(v) are givenin (4.100), (4.101), and (A.51), respectively. v1, v2, and v? are chosen to satisfyQc(P2,1(t)

)= ε, Qc

(P2,2(t)

)= ε and Qc

(P?

2 (t))= ε, respectively.

Proof. The suboptimal strategy I is derived based on two simplifications of the optimalstrategy. Note that the new notation of the Lagrangian variable v is used instead ofv in order to indicate a different value of the Lagrangian variable in the strategy.Specifically, we have

1. Linearization of the nonlinear function x2(t, v):

Given certain v, we expand F(t) =√

1− 4v + 4avt using the first order Taylorexpansion at v/a as

F(t) ≈ F(v

a

)+ F

′(v

a

)(t− v

a

)= 1− 2v +

2av1− 2v

(t− v

a

). (4.112)

where 1− 2v > 0 holds in order to guarantee the positivity of x2(t, v), cf. Ap-pendix A.10. Applying (4.112) in the expression of x2(t, v) in (4.99) yields x2(t, v)in (4.111). Consequently, the term x2(t, v) in (4.102) and (4.105) is then replacedby x2(t, v) in (4.106), (4.109), and (4.110).

2. Simplification of the optimization problem (4.104):

In order to avoid solving the optimization problem (4.104), we select v? from twovalues v1 and v2 corresponding to two extreme choices of the discontinuities.More particularly, recalling that for the power allocation strategy (4.105), thefeasible domain of γ(v) is γ(v) ∈ [T3(v), T1(v)], we only consider the two casesthat γ(v) is located at the boundary points. If we choose the discontinuity atT1 (v), v1 is chosen to let the constraint Qc

(P2,1(t)

)= ε be satisfied and the

solution P2,1(t) in (4.109) is obtained. If we choose the discontinuity at T3 (v?),the solution is given in (4.110) with v2 chosen to satisfy Qc

(P2,2(t)

)= ε.

Integrating these two steps in the optimal strategy yields the form of the suboptimalstrategy I.

In order to determine v? in the third case of Proposition 4.3.2, we need to evaluatethe influence of v on Qc(P2(t)) where the relations between P2(t) and v are given in(4.106), (4.107), (4.109), or (4.110) for the corresponding regions of v. In contrast tothe numerical computation of Qc(P2(t)) in the optimal strategy, we exploit the linearstructure of the suboptimal strategy I and derive the analytical form of Qc(P2(t)). Fornotational simplicity, the metric Qc(v) = Qc(P2(t)) is used in the following proposi-tion which explicitly shows its dependence on v.

Proposition 4.3.3. The function Qc(v) resulting from the suboptimal solution P?2 is as follows

• If 1− 2v ≥ 2avPS:

Qc(v) =1

1 + aPSQ1 (T1 (v)) +

1− 2va

(Q2 (ζ, T2 (v))−Q2 (ζ, T1 (v)))

+ (1−Q1 (T2 (v))) . (4.113)


• If 1− 2v ≤ 0:

Qc(v) = 1− aPS

1 + aPSQ1 (T2 (v)) . (4.114)

• If 0 < 1− 2v < 2avPS:

a) Using the power allocation strategy in (4.109), we have

• If 0 < 1− 2v ≤ avPS: P2(t) reduces to a switching between 0 and PS at T1(v)

Qc (v) = 1− aPS

1 + aPSQ1 (T1 (v)) . (4.115)

• If avPS < 1− 2v < 2avPS: the power allocation strategy P2(t) in (4.109) has thesame form as P2(t) in (4.106), thus Qc(v) has the same form as (4.113).

b) Using the power allocation strategy in (4.110), we have

Qc(v) =1

1 + aPSQ1 (T3 (v)) +

1− 2va

(Q2 (ζ, T2 (v))−Q2 (ζ, T3 (v)))

+ (1−Q1 (T2 (v))) . (4.116)

In (4.113)-(4.116), we use the variable ζ = (1− v)/a. The functions Q1(T) and Q2(λ, T)are defined in (A.72) and (A.73).


Given the power allocation strategy in Proposition 4.3.2, we derive a closed-formexpression of the achievable performance. The results are summarized as follows.

Proposition 4.3.4. The achievable rate R(P?

2 (t))

of the suboptimal strategy I is

1. R(P?

2 (t))= limT→∞ r (T, 1, PS), if 0 ≤ ε ≤ 1/(1 + aPS). This limit can be straight-

forwardly obtained from (4.24).

2. R(P?

2 (t))= 0, if ε = 1, which follows directly with P?

2 (t) = 0, ∀t.

3. If 1/(1 + aPS) ≤ ε ≤ 1, we distinguish among three cases.

• 1− 2v? ≥ 2av?PS:

R(P?

2 (t))= r (T1 (v?) , 1, PS) + r (T2 (v?) , η, θ)− r (T1 (v?) , η, θ) . (4.117)

• 1− 2v? ≤ 0:R(P?

2 (t))= r (T2 (v?) , 1, PS) . (4.118)

• 0 < 1− 2v? < 2av?PS:a) If the selection problem (4.104) results in P?

2 (t) = P2,1(t) and v? = v1, then wehave


• If 0 < 1− 2v? ≤ av?PS, then

R(P?

2 (t))= r (T1 (v?) , 1, PS) . (4.119)

• If av?PS < 1− 2v? < 2av?PS, then R(P?

2 (t))

has the same form as (4.117).

b) If the selection problem (4.104) yields P?2 (t) = P2,2(t) and v? = v2, then we

have

R(P?

2 (t))= r (T3 (v?) , 1, PS) + r (T2 (v?) , η, θ)− r (T3 (v?) , η, θ) (4.120)

In above, we use the variables

η = − 2v?

1− 2v?, θ =

v?

(1− 2v?) a. (4.121)


4.3.2.2 Suboptimal Power Allocation Strategy II

In general, convex optimization problems provide favorable properties in the designof the optimal strategy [111]. Motivated by this fact, we approximate the primal non-convex problem by a convex relaxation and develop the suboptimal strategy II basedon the newly-formed convex problem. Specifically, applying Jensen’s inequality toQc (P2(t)), we obtain its lower bound as

Qc (P2(t)) ≥1

1 + aEt{P2(t)}. (4.122)

Consequently, the non-convex constraint (4.91) is approximated by a convex constraint

Et{P2(t)} ≤1a

(1ε− 1)

. (4.123)

Replacing the second constraint in (4.93) with (4.123) yields

maxP2(t)

Et

{ln(

1 +P2(t)

t

)}(4.124)

s.t. 0 ≤ P2(t) ≤ PS, ∀t ≥ 0(4.123).

which is a convex optimization problem. Compared to (4.93), the problem (4.124)aims at satisfying a stricter constraint than the primal constraint, yielding a moreconservative power allocation strategy. Considering that the constraint (4.123) is in


the form of an average IT constraint, the optimization problem (4.124) is thus solvedby the algorithm in Section 4.2.1. The solution is given as

P?2 (t) =

PS, if 0 ≤ t < v? − PS

0, if t > v?

v? − t, otherwise

(4.125)

where v? is chosen to satisfy the constraint (4.123).The achievable performance of (4.125) is given as

R (P?2 (t)) = r (v? − PS, 1, PS) + r (v?, 0, v?)− r (v? − PS, 0, v?) (4.126)

where r(T, α, β) is given in (4.24).The simulation parameters are given as follows unless explicitly stated. Assuming

Rayleigh fading channels for all links, the channel power gain of each fading linkgij, i, j = 1, 2, follows an exponential distribution with l11 = 1, l21 = 2, l12 = 3 andl22 = 1. The noise variances of the primary and secondary links are σ2

P = 1 andσ2

S = 1, respectively. The primary transmit power is P1 = 10 dB. In the following, weevaluate the achievable rate in the unit bits/s/Hz.

The performance of four power allocation strategies is compared:

• The optimal strategy given in Theorem 4.3.1.

• The suboptimal strategy I given in Proposition 4.3.2.

• The suboptimal strategy II given in (4.125).

• Non-adaptive power transmission such that the power value is chosen to satisfyall constraints in (4.93) at every channel realization [93].

Pc(t) = min{

PS,1a

(1ε− 1)}

. (4.127)

Figure 4.8 depicts the achievable secondary rate R versus the peak transmit powerlimit PS with different values of the desired outage rate γp. The pre-defined outageprobability limit is εp = 20%. We have the following observations. Firstly, all pro-posed strategies outperform the non-adaptive power transmission strategy beyondsome value of PS. The reason is that for a small value of PS, all four strategies result inthe same optimal solution P?

2 (t) = PS, ∀t, when the outage constraint is satisfied, i.e.,yielding the same achievable rate. For a large value of PS, non-adaptive power trans-mission aims at satisfying all constraints for each channel realization, while othersaim at exploiting the channel variations to adapt the power. Secondly, the suboptimalstrategy I is near-optimal and outperforms the second suboptimal strategy beyondsome value of PS. The performance gain reflects the advantage of exploiting the pri-mary CSI over the conventional IT constraint. The reason is as follows. If equality in(4.122) holds, then both the primal constraint (4.91) and the approximated constraint


0 5 10 15 20 25 30 34

0.5

1

1.5

2

2.5

3

Transmit power constraint PS [dB]

Ach

ieva

ble

rate

R[b

its/

s/H

z]

γp = 0.2bits/s/Hzγp= 0.6bits/s/Hzγp= 1.0bits/s/Hzγp= 1.4bits/s/Hz

Figure 4.8: Achievable rate R versus PS for different primary outage rate values γpwith the outage probability limit is εp = 20%. The lines/markers ofthe same color share the same setup parameter γp. Dash-dotted lines:optimal solution (sim.); solid lines: suboptimal solution I (sim.); circles:suboptimal solution I (analytical); dashed lines: suboptimal solution II(sim.); triangles: suboptimal solution II (analytical); dotted lines: non-adaptive power transmission (sim.).

(4.123) are equivalent, resulting in no performance loss of the suboptimal strategy II.This case happens when P?

2 (t) = PS, ∀t, for a small value of PS. For a large valueof PS, P?

2 (t) depends on the channel variation t, i.e., (4.122) is satisfied with strict in-equality. Consequently, there exists a performance loss of the suboptimal strategy II.Thirdly, the theoretic performance analysis matches the numerical results for the twosuboptimal strategies. Finally, we note that with an increase of PS, the performance ofthe optimal and suboptimal strategy I increases first, stays constantly later, and thenincreases afterwards. This observation is different from that of the suboptimal strat-egy II in which the performance saturates beyond a certain value of PS. The reasonis roughly explained as follows. For the optimal solution in Theorem 4.3.1, v? de-creases with an increase of PS given a fixed γp and εp in the outage constraint. Thus,T1 (v?) in (4.100) can be negative for some values of PS yielding the condition to letP?

2 (t) = PS never satisfied, cf. (4.95). Under such cases, the performance remains con-stant with an increase of PS since the peak power constraint is not active. However,since T1 (v?) is a quadratic form of PS and a linear function of v?, it is possible that anfurther increase of PS yields a positive T1 (v?) and consequently an active peak powerconstraint. Hence, the achievable rate increases with an increase of PS. Similar expla-nation is applied to the suboptimal solution I. For the suboptimal solution II, v? in(4.125) decreases with an increase of PS. Once v? is smaller than PS, increasing PS fur-

4.4. Summary 79

0 5 10 15 20 25 30 34

0.5

1

1.5

2

2.5


Ach

ieva

ble

rate

R[b

its/

s/H

z]

εp= 15 %εp= 20 %εp= 25 %εp= 30 %

Figure 4.9: Achievable rate R versus PS for different outage probability limits εp; theoutage primary rate is γp = 1bits/s/Hz. The explanations of the linesand markers are the same as in Figure 4.8. The lines/markers of thesame color share the same setup parameter εp.

ther leads the condition of P?(t) = PS always be satisfied, cf. (4.125). In other words,its performance is invariant to an increase of PS beyond some limit. The achievablerate R is plotted versus PS under different outage probability thresholds εp in Figure4.9. The desired outage capacity is γp = 1 bits/s/Hz. We observe similar trends as inFigure 4.8 for different outage probability requirements.

In order to further visualize reasons of the performance gain of the optimal andthe first suboptimal strategies over the second one for a large value of PS, Figure 4.10illustrates the resulting outage probability of proposed strategies with different outageprobability limits εp. The outage capacity is set to γp = 1 bits/s/Hz as in Figure4.9. For a small value of PS, all proposed strategies yield the same solution P?

2 (t) =PS, ∀t, thus the same outage probabilities. For a large value of PS, the suboptimalstrategy II yields a more conservative method to protect the PU, i.e., a lower outageprobability than εp. On the contrary, the outage probability caused by other proposedstrategies is shown to meet the requirement. Combining the results shown in Figure4.9, we conclude that a performance gain w.r.t. secondary achievable rate is achievedat the expense of a larger primary outage probability of the the optimal and the firstsuboptimal method.

4.4 Summary

The goal in the design of spectrum sharing systems is to reduce the performancedegradation to the primary transmission by adjusting the secondary transmission.


0 5 10 15 20 25 30 340.14

0.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3


Out

age

Prob

abili

ty

εp= 15 %εp= 20 %εp= 25 %εp= 30 %

Figure 4.10: Resulting outage probability versus PS for different εp; the outage pri-mary rate is γp = 1bits/s/Hz. The lines/markers of the same colorshare the same setup parameter εp. lines with squares: optimal so-lution; lines with circles: suboptimal solution I; lines with triangles:suboptimal solution II.

Due to limited cooperation between the primary and secondary systems, the SUsmay only have partial CSI related to the primary system in reality. Hence, the designof the optimal transmission strategy and the derivation of the achievable performanceof the secondary transmission under such a challenge is imperative.

In this chapter, we have addressed this issue by studying the power allocation ofa single-antenna ST aiming at maximizing the achievable rate of the secondary linksubject to different QoS constraints of the primary link. Due to realistic restrictions, weassumed the ST has only partial CSI related to PR. Both optimal and computationallyefficient suboptimal solutions were developed. Furthermore, the performance of thesuboptimal strategies was derived in closed form. Through performance evaluations,the near-optimality of the low-complexity suboptimal solutions was validated. Thus,the achievable performance of the system can be approximately assessed. Moreover,we verified the advantage of additionally exploiting statistical CSI of the primarylinks by applying the outage probability constraint instead of the IT constraint on theprimary transmission. The performance gain can also be approximately evaluatedby the derived analytical results. Overall, we have contributed to the quantitativeassessment of the achievable performance of the SUs at the expense of a performancedegradation at the PUs by exploiting different amounts of available partial CSI. Suchresults not only characterize the benefit in exploring the spectrum sharing paradigm,but also facilitate the decision in choosing power adaptation strategies.

Chapter 5

Robust Transceiver Optimization

The use of multiple antennas at the transceivers allows for the enhancement of thespectral efficiency by exploiting the spatial DoFs [114,131,135]. Specifically, the signalvector is transmitted towards the desired user with the optimized precoders, whileMUI and the effect of fading channels are mitigated at the receiver with the optimizedequalizers. Recently, this strategy has been applied to cognitive downlink transmis-sion [60,143]. It effectively improves the spectral utilization and limits the interference.

Due to imperfect or outdated channel estimates and the quantization effect in thelimited feedback, erroneous CSI is inevitable. Moreover, in cognitive networks, thelimited cooperation between the PUs and the SUs brings new challenges to obtainperfect CSI related to the primary link. Since the presence of imperfect CSI severelydegrades the achievable performance [96], or even causes the violation of the inter-ference power limit to the PU, robust design becomes indispensable to deal with thechannel uncertainties.

As introduced in Chapter 2, two types of CSI error models are commonly consid-ered: the bounded and the stochastic model. Two corresponding principles of robustdesign are adopted. One is the worst-case approach to guarantee a certain system per-formance for any channel realization within the CSI uncertainty region [35, 144, 153].The other is the stochastic approach which guarantees a certain system performanceaveraged over all channel realizations within the uncertainty region [151].

This chapter considers the infrastructure-based cognitive downlink transmissionwhere one secondary BS and multiple SUs coexist with the PUs. We target at de-signing the robust transceiver filters considering both CSI error models: the boundedCSI error in Section 5.1 and the stochastic CSI error in Section 5.2. To ease the com-putational burden at the SUs, the optimization of both precoders and equalizers isperformed at the BS and the equalizers are fed forward to each receiver afterwards.Alternatively, the task of the equalizer optimization can also be assigned to each re-ceiver if it has perfect information of the channel coefficient.

The results presented in this chapter have been published in part by the authorin [42–44, 50]1.

1 In reference to IEEE copyrighted material which is used with permission in this thesis, the IEEEdoes not endorse any of RWTH Aachen University’s products or services. Internal or personal useof this material is permitted. If interested in reprinting/republishing IEEE copyrighted material foradvertising or promotional purposes or for creating new collective works for resale or redistribution,please go to http://www.ieee.org/publications_standards/publications/rights/rights_link.html to learn how to obtain a License from RightsLink.

81



82 Chapter 5. Robust Transceiver Optimization

Figure 5.1: Spectrum sharing of the cognitive network with K SUs and P PUs.

5.1 Bounded CSI Error

We consider the downlink CR network depicted in Figure 5.1 where one secondaryBS communicating with K SUs. The number of transmit antennas at the BS is given byNt. Let Nk indicate the number of data streams for the kth user and sk = [s1

k, . . . , sNkk ]T

be the symbol vector intended to the kth user. Stacking data vectors into one N × 1global vector, the data stream for all SUs is s = [sT

1 , . . . , sTK]

T with N =∑K

k=1 Nk. Weassume

E{ssH} = IN.

The precoding matrix for the kth SU is represented by Gk ∈ CNt×Nk which yields theprecoding matrix at the secondary BS as G = [G1, . . . , GK]. The transmit vector at thesecondary BS is x = Gs. Consequently, the average transmitted power at the BS is

E[‖x‖2

]= tr

{GGH

}= ‖G‖2

F . (5.1)

Assuming the kth SU is equipped with Lk antennas, the downlink flat fading channelfrom the BS to the kth user is given by Hk ∈ CLk×Nt . At the receiver side, the linearequalizer Fk is performed to obtain the recovered data stream sk ∈ CNk×1

sk = Fk (HkGs + ωk) , ∀ k = 1, . . . , K (5.2)

where ωk ∼ CN (0, σ2k I) indicates the noise at the kth SU with the variance σ2

k .

5.1. Bounded CSI Error 83

There are P PUs coexisting with the cognitive downlink transmission. The pth PUis equipped with Lp antennas, ∀ p = 1, . . . , P. Denoting the interfering link from thesecondary BS to the pth PU as Yp ∈ CLp×Nt , the received interference signal at the pthPU from the secondary BS is zp = YpGs with the interference power given by

E[∥∥zp

∥∥2]=∥∥YpG

∥∥2F , ∀ p = 1, . . . , P (5.3)

We use mean squared error (MSE) as the performance measures. The MSE metricbetween the symbol vector sk and the recovered symbol sk for the kth user is definedas

MSEk = E{‖sk − sk‖2

2

}= ‖FkHkG−Qk‖2

F + σ2k ‖Fk‖2

F (5.4)

where

Qk =[0Nk×

∑k−1l=1 Nl

, INk , 0Nk×∑K

l=k+1 Nl

]. (5.5)

The comparison between the MSE measure to other performance measures such asbit error rate (BER) and SINR are elaborated in several works, e.g., [30, 98]. Brieflyspeaking, the optimization problem based on the BER measure is usually more diffi-cult to handle [98]. Moreover, from the estimation perspective, considering the MSEas a performance measure may be more informative than SINR since it directly re-veals the difference between the transmit and recovered signals [30]. Nevertheless,MSE measure is strongly connected to SINR or BER measures [98, 99].

Considering the channel uncertainties, the channel matrix Hk, ∀ k = 1, . . . , K isthe superposition of two parts

Hk = Hk + ∆Hk (5.6)

where Hk denotes either the channel estimate or the feedback CSI at the BS and theerror matrix of Hk is given by ∆Hk. The similar definition is applicable for the channelmatrix Yp

Yp = Yp + ∆Yp (5.7)

with Yp as the channel estimate or the feedback CSI at the secondary BS and theerror matrix of Yp is given by ∆Yp, ∀ p = 1, . . . , P. According to the ellipsoidaluncertainty model [153], the imperfect CSI Hk and Yp are within the region Hk andYp, respectively

Hk ={

Hk|Hk = Hk + ∆Hk, tr(

∆HkCk∆HHk

)≤ ε2

k

}(5.8)

Yp ={

Yp|Yp = Yp + ∆Yp, tr(

∆YpDp∆YHp

)≤ ζ2

p

}(5.9)

where a priori known matrices Ck and Dp are positive definite Hermitian matrices thatdefine the shape of the uncertainty regions. The sizes of the uncertainty region of Hkand Yp are quantized by εk and ζp, respectively, k = 1, . . . , K and p = 1, . . . , P.


We aim at minimizing the sum-MSE of all secondary links subject to the trans-mit power constraint P0 at the secondary BS and interference power constraint Ppreceived at the PR. Assuming only erroneous CSI is available at the secondary BS, theprecoding matrices G and equalizer matrices F1, . . . , FK need be designed to ensurethe performance targets of both the PUs and the SUs is guaranteed for all channel re-alizations within the uncertainty region through the optimization. The optimizationproblem is formulated as

minG,F1,...,FK

maxHk∈Hk

K∑k=1

MSEk (5.10)

s.t. ‖G‖2F ≤ P0

maxYp∈Yp

∥∥YpG∥∥2

F ≤ Pp, p = 1, . . . , P.

Although only the sum-MSE metric is considered here, the algorithm design could bestraightforwardly extended to the problems targeted at other MSE-based measures.

The problem (5.10) generalizes the non-convex problem in [131] by extending theCSI uncertainty model from the ball region to the ellipsoidal region and additionallyconsidering the interference constraint encountered in the CR network. Thus, it is alsoa non-convex optimization problem, and moreover, a semi-infinite problem which isgenerally intractable [144]. In order to solve this problem, the authors in [131] pro-vided a framework by applying an alternating principle to iteratively optimize the

Algorithm 1 Alternating algorithm for robust optimization with bounded CSI

Initialization: Iteration number index l ← 0, maximum allowable iterations lmax ←Lmax, the desired accuracy ξ, the initial receivers F(0)

k for all ∀ k = 1, . . . , K.repeat

l ← l + 1Update the precoder G(l) with fixed equalizers F(l−1)

k , k = 1, . . . , K{MSE

(l)k

k=1,...,K, G(l)

}← min

Gmax

Hk∈Hk

K∑k=1

MSE(l−1)k

s.t.: ‖G‖2F ≤ P0 (5.11)

maxYp∈Yp

∥∥YpG∥∥2

F ≤ Pp, p = 1, . . . , P.

Update the equalizers F(l)k , k = 1, . . . , K with fixed precoder G(l):{

MSE(l)k , F(l)

k

}← min

Fkmax

Hk∈HkMSE

(l)k (5.12)

until l ≥ Lmax OR the MSE target converges with the precision ξ


precoders and the equalizers. In this section, analogy to [131], we apply the alternat-ing principle to (5.10). Particularly, decomposing (5.10) into two subproblems (5.11)and (5.12) which tackles with the optimization of the precoders and the equalizers,respectively, the iterative algorithm is then proposed in Algorithm 1. In the following,we seek for the efficient algorithm to solve each subproblem. The convergence behav-ior of Algorithm 1 is also shown based on the convexity of each subproblem. For thesake of brevity, the iteration number index is omitted in the following derivation.

First, we reformulate the subproblems (5.11) and (5.12) using the vectorizationproperty as

vec(ABC) =(

CT ⊗A)

vec(B), (5.13)

the MSE term of kth user (5.4) can be reformulated to

MSEk = ‖ψk + Ψkδk‖22 + σ2

k ‖Fk‖2F (5.14)

where

ψk(G, Fk) = vec(FkHkG−Qk) (5.15)

Ψk(G, Fk) = GT ⊗ Fk (5.16)δk = vec(∆Hk). (5.17)

Moreover, the interference power at the pth PU is reformulated as

∥∥(Yp + ∆Yp)G∥∥2

F =∥∥∥φp + Φpδp

∥∥∥2

2(5.18)

with

φp = vec(YpG) (5.19)

Φp = GT ⊗ I (5.20)δp = vec(∆Yp). (5.21)

Similarly, the uncertainty regions Hk and Yp in (5.8) and (5.9) are converted to

Hk ={

Hk = Hk + ∆Hk|∥∥∆HkCk

∥∥F =

∥∥∥(CTk ⊗ I)δk

∥∥∥2≤ εk

}(5.22)

Yp ={

Yp = Yp + ∆Yp|∥∥∆YpDp

∥∥F =

∥∥∥(DTp ⊗ I)δp

∥∥∥2≤ ζp

}(5.23)

respectively, where we use the matrix decomposition

Ck = CkCHk (5.24)

Dp = DpDHp (5.25)

and the vectorization in (5.13).


Inserting (5.14) and (5.18) into the subproblem (5.11), we have

minG,{τk}K

k=1

maxHk∈Hk

K∑k=1

τk (5.26)

s.t. ‖ψk + Ψkδk‖22 ≤ τk

maxYp∈Yp

∥∥∥φp + Φpδp

∥∥∥2

2≤ Pp, ∀ p = 1, . . . , P

‖G‖2F ≤ P0

τk ≥ 0 ∀ k = 1, . . . , K

where τk is an auxiliary slack variable. The term∑K

k=1 σ2k ‖Fk‖2

F is omitted due to thefixed value of Fk, ∀ k = 1, . . . , K. The subproblem (5.12) is rewritten as

minFk, {τk}K

k=1

maxHk∈Hk

τk + σ2k ‖Fk‖2

F (5.27)


τk ≥ 0 ∀ k = 1, . . . , K

The following lemma is useful to reformulate the subproblems into the form of stan-dard convex optimization problem.

Lemma 5.1.1. ( [29, Lemma. 2]) Let A be a Hermitian matrix. Then

A � BHDC + CHDHB, ∀ D : ‖D‖2 ≤ ε (5.28)

if and only if there exists a λ ≥ 0 such that[A− λCHC −εBH

−εBH λI

]� 0. (5.29)

Considering the first subproblem (5.26), if we ignore the interference power con-straint first, the following result is obtained.

Proposition 5.1.1. Dropping the interference power constraint, (5.26) is rewritten in thefollowing semi-definite programming (SDP) form as

minG,{τk}K

k=1,{λk}Kk=1

K∑k=1

τk (5.30)

s.t.

τk − λk ψHk 0

ψk I −εkΨk

0 −εkΨHk λkI

� 0

τk ≥ 0, λk ≥ 0, ∀ k = 1 . . . , K‖G‖F ≤ P0,


with Ψk =(

C−1k G

)T⊗ Fk.

Proof. Ignoring the interference power constraint, the optimization problem (5.26) isreformulated into

minG,τ1,...,τK

maxHk∈Hk

K∑k=1

τk (5.31)


‖G‖F ≤ P0.τk ≥ 0 ∀ k = 1, . . . , K.

Applying the Schur Complement Theorem [56] to the first constraint in (5.31), it yields[τk ψH

kψk IK

]+

[0 (Ψkδk)

H

Ψkδk 0

]� 0. (5.32)

We define the matrix

Ψk = Ψk

(CT

k ⊗ I)−1

(a)=(

GT ⊗ Fk

)(C−T

k ⊗ I)

(b)=(

C−1k G

)T⊗ Fk (5.33)

where, in (a), we use the property of the Kronecker product that

(A⊗ B)−1 = A−1 ⊗ B−1.

In (b), we use(A⊗ B) (C⊗D) = (AC⊗ BD) .

Using (5.33), we construct the following matrices

A =

[τk ψH

kψk I

]B =

[0 ΨH

k

]C =

[−1 0

]D =

(CT

k ⊗ I)

δk

ε = εk.

and then the constraint (5.32) can be written in the form of

A � BHDC + CHDHB.


According to Lemma 5.1.1, this condition subject to the uncertainty region (5.22) canbe reformulated as τk − λk ψH

k 0ψk I −εkΨk

0 −εkΨHk λkI

� 0.

Consequently, the problem (5.31) is converted to (5.30).

Analogously, considering the interference power constraint of the pth PU in (5.26),we insert the following matrices in (5.29)

A =

[Pp φH

p

φp I

], B =

[0 ΦH

p

]C =

[−1 0

], D = vec

((DT

p ⊗ I)

δp

), ε = ζp.

withΦp =

(D−1

p G)T⊗ I (5.34)

and obtain the equivalent form of the interference power constraint within the uncer-tainty region is

Pp − ηp φHp 0

φp I −ζpΦp

0 −ζpΦHp ηpI

� 0, (5.35)

with ηp ≥ 0, ∀ p = 1, . . . , P.Combining (5.30) and (5.35), the subproblem (5.26) is represented as

minG,{λk}K

k=1,{τk}Kk=1,{ηp}P

p=1

K∑k=1

τk (5.36)

s.t.

τk − λk ψH

k 0

ψk I −εkΨk

0 −εkΨHk λk I

� 0

Pp − ηp φH

p 0

φp I −ζpΦp

0 −ζpΦHp ηpI

� 0

‖G‖2F ≤ P0, ∀k

τk ≥ 0, λk ≥ 0, ∀ k = 1, . . . , Kηp ≥ 0, ∀ p = 1, . . . , P. (5.37)


Next, considering the subproblem (5.12), it is similar to the equalizer optimizationin [131] since the additional interference constraint does not affect given the precodingmatrix G fixed. Therefore, the derivation in [131] can be directly applied except theextension from the ball uncertainty model to the ellipsoidal model. The equivalentSDP form of subproblem (5.27) is

minFk,{λk}K

k=1,{τk}Kk=1

τk + σ2k ‖Fk‖2

F (5.38)

s.t.

τk − λk ψH

k 0

ψk I −εkΨk

0 −εkΨHk λkI

� 0.

λk ≥ 0, τk ≥ 0, ∀ k = 1, . . . , K

So far, the subproblems in Algorithm 1 is reformulated into the convex SDPform in (5.36) and (5.38) which can be efficiently solved by convex optimization tool-boxes [24], [82]. The implementation complexity of the SDP problem is polynomialin the problem size and ln(1/ξ) where ξ is the required accuracy [88]. Concern-ing the optimality issue, we show that Algorithm 1 is guaranteed to converge to thelocal optimum for any initial selection of equalizers. In order to prove it, first wenote that (5.11) is feasible for any initialized equalizers. Assuming MSE(1a) and G(1)

are the sum-MSE and the precoder obtained in the first iteration by solving (5.11),then MSE(1a) is also one feasible solution for (5.12). Assuming the subsequent so-lution of (5.12) is given by MSE(1b), it can only be smaller or equal to MSE(1a), i.e.,MSE(1b) ≤ MSE(1a). In the second iteration, since G(1) is always in the feasible set.so the problem (5.11) will find the precoder G(2) yielding smaller or equal MSE, i.e.,MSE(2a) ≤ MSE(1b). The rest of the iterative process is deduced by analogy. Therefore,Algorithm 1 creates a monotonically non-increasing sequence of the MSE target withthe lower-bound equal to zero. This indicates its convergence to the local optimum.However, the global optimum is not guaranteed.

To evaluate the performance, we consider the CR network consisting of one sec-ondary BS with Nt = 6 transmit antennas and two SUs. The simulation parametersfor two SUs are the same. Each SU is equipped with Lk = 2 antennas and transmitsNk = 2 data streams. One PU with a single antenna coexists with the secondarydownlink transmission. Concerning the CSI error model in (5.22) and (5.23), theshape parameters of the ellipsoidal region Ck and Dp are all identity matrices. Theerror bound parameters εk and ζp are set to 0.1. All power values in dB are givenrelative to the reference level of value 1. The transmit power at the secondary BS isbounded by P0 = 0 dB and the interference power received at the PU is constrainedby Pp = −10 dB. The desired accuracy ξ in simulations is 10−4.

Four transceiver strategies are compare:

• Perfect CSI: The strategy is in Algorithm 1 assuming no channel uncertainty.

• Robust: The strategy is in Algorithm 1.


0 5 10 15 20 25 30 35 4010−3

10−2

10−1

100

SNR [dB]

Sum

-MSE

Perfect CSIRobustRobust w. MMSE rec.Non-robust

Figure 5.2: Sum-MSE vs. SNR of single CR link.

−12 −11 −10 −9 −8 −7 −6

0.2

0.4

0.6

0.8

Interference Power [dB]

CD

F

RobustNon-robust

Figure 5.3: CDF of the worst-case interference power at the PR.

5.2. Stochastic CSI Error 91

• Robust w. minimum mean squared error (MMSE) rec.: the strategy is given in Al-gorithm 1 and the equalizers are calculated according to MMSE criterion after-wards based on the optimal precoder and the worst-case channel coefficients.Similar to [131, Appendix. A], the worst-case channel is obtained which yieldsthe maximum sum-MSE.

• Non-robust: the transceiver optimization are calculated by the strategy in Algo-rithm 1 and neglecting the error in the estimated channel matrices.

In Figure 5.2, we plot the result of sum-MSE versus the SNR of each cognitive link.The proposed robust solution is shown to provide marginal performance gain com-paring to the non-robust solution. With the MMSE receiver additionally calculatedcorresponding to the worst-case channel, the MSE target can be further reduced. Weremark that the interference power constraint is often violated in the non-robust case,as shown in Figure 5.3. In the cognitive network, the infringement of the interferencepower may cause the severe performance degradation to the PU, which should bestrictly refrained.

5.2 Stochastic CSI Error

In Section 5.1, we investigate the robust transceiver design with the bounded CSI errorand use the worst-case principle. However, it is usually over-conservative from thesystem perspective. In the following, we address the robust design based on stochasticCSI error and apply the stochastic principle.

We consider a multiple-input multiple-output (MIMO) downlink CR network inFigure 5.1. The input-output relations of the system are identical to that in Section5.1. The difference is in the CSI error model. Specifically, according to the Kroneckermodel [68], the channel from the secondary BS to the kth secondary user follows thedistribution as

vec(Hk) ∼ CN(

0, RRxk ⊗ RTx

), ∀ k = 1, . . . , K

where RRxHk

and RTx are the receive and transmit correlation coefficient matrices. Byperforming MMSE channel estimation [95], the channel is expressed as

Hk = Hk + ∆Hk (5.39)

where Hk is the estimated channel matrix. The corresponding channel estimationerror matrix ∆Hk follows as

vec (∆Hk) ∼ CN(

0, Re,RxHk⊗ Re,Tx

Hk

). (5.40)

with Re,RxHk

and Re,TxHk

given as the row and column covariance matrices of ∆Hk, respec-tively. In general, they can be expressed as a function of RRx

Hkand RTx depending on

the specific channel estimation method [27,95]. A special case that the covariance ma-


trices RRxHk

and RTx are the scaled identity matrices has been studied in the supervisedthesis [59].

A similar channel model applies to the channel matrix Yp from the secondaryBS to the pth PU. The corresponding estimated CSI and channel error matrices aredenoted by Yp and ∆Yp, ∀ p = 1, . . . , P

Yp = Yp + ∆Yp (5.41)

vec(∆Yp) ∼ CN(

0, Re,RxYp⊗ Re,Tx

Yp

)The optimization problem aims at minimizing the sum-MSE of the secondary net-

work subject to the transmit power constraint P0 and the interference power constraintPp at the pth PU

minG,F1,...,FK

K∑k=1

EHk {MSEk} (5.42)

s.t. ‖G‖2F ≤ P0

EYp

{∥∥YpG∥∥2

F

}≤ Pp, ∀ p = 1, . . . , P.

In order to reveal the convexity of (5.42), we need to express the objective function andthe interference constraint as explicit functions of the optimization variables. Specifi-cally, we integrate the stochastic channel model (5.39) and apply [151, Lemma. 1], tothe objective function in (5.42), it yields

EHk {MSEk}

=∥∥FkHkG−Qk

∥∥2F + σ2

k ‖Fk‖2F

+ tr

{tr

{ K∑k=1

GkGHk

(Re,Tx

Hk

)T}‖Fk‖2

F Re,RxHk

}

= tr

{INk − FkHkGk −GH

k HHk FH

k + σ2k FkFH

k

+K∑

l=1

Fk

(HkGlGH

l HHk + tr

(GlGH

l

(Re,Tx

Hk

)T)

Re,RxHk

)FH

k

}. (5.43)

where in (5.43), we use the definition of Qk in (5.5). Analogously, by incorporating(5.41) and applying [151, Lemma. 1], the interference power constraint for the pth PU,∀ p = 1, . . . , P, is rewritten to

EYp

{∥∥YpG∥∥2

F

}= tr

{ApGGH

}(5.44)

withAp = YH

p Yp + tr{

Re,RxYp

}(Re,Tx

Yp

)T. (5.45)


Algorithm 2 SOCP-based robust design

Initialization: Iteration number index l ← 0, maximum allowable iterations lmax ←Lmax, the desired accuracy ξ, initial receivers F(0)

k for all ∀ k = 1, . . . , K.repeat

l ← l + 1Update the precoder G(l) with fixed equalizers F(l−1)

k , ∀ k = 1, . . . , K.{ K∑k=1

EHk

{MSE(l)

k

}, G(l)

}← min

G(l)

K∑k=1

EHk

{MSE(l−1)

k

}(5.46)

s.t. ‖G(l)‖2F ≤ P0,

EYp

{∥∥∥YpG(l)∥∥∥2

F

}≤ Pp, ∀p = 1, . . . , P.

Update the equalizers F(l)k , ∀ k = 1, . . . , K, with fixed precoder G(l)

{MSE(l)

k , F(l)k

}← min

F(l)k

EHk

{MSE(l)

k

}(5.47)

until l ≥ Lmax OR the MSE target converges with the precision ξ.

As shown in (5.43), the objective function of (5.42) is non-convex with the joint op-timization of the precoders and equalizers which yielding (5.42) to be a non-convexoptimization problem. Generally, the optimal solution of such problem is difficult toobtain. In what follows, we design three efficient algorithms to solve it.

5.2.1 SOCP-based Algorithm

Similar to Section 5.1, we apply the alternating principle to solve the optimizationproblem (5.42). In particular, we first decompose (5.42) into two subproblems (5.46)and (5.47) to compute the precoders and the equalizers separately, and propose an al-gorithm to solve two subproblems iteratively. The structure of this strategy is outlinedin Algorithm 2 named second-order cone programming (SOCP)-based algorithm.

In the current form of (5.46) and (5.47), the convexity of each subproblem isunclear. In the following, we address the reformulation into the convex optimiza-tion form. Considering the precoder design (5.46) in which the objective functionis expressed in (5.43), the term σ2

k ‖Fk‖2F in (5.43) remains constant due to the fixed

equalizers, thus, can be omitted. By introducing the auxiliary variables τk and rk,∀ k = 1, . . . , K, (5.46) is rewritten as

minG,{τk}K

k=1,{rk}Kk=1

K∑k=1

(τk + tr

{rkFkRe,Rx

HkFH

k

})(5.48)

s.t. ‖FkHkG−Qk‖F ≤√

τk,


‖G‖F ≤ PT,∥∥∥(Re,TxHk

)T/2G∥∥∥

F≤ rk

τk ≥ 0, ∀ k = 1, . . . , Krk ≥ 0, ∀ k = 1, . . . , K∥∥∥A1/2

p G∥∥∥

F≤ Pp, ∀ p = 1, . . . , P.

For the equalizer design in (5.47), the objective function is convex w.r.t. Fk andindependent from Fi, ∀ i = 1, . . . , K and i 6= k. Therefore, the equalizer of each SU isoptimized independently. The closed-form optimum for the kth user F?

k is obtainedby calculating the derivative of (5.43) w.r.t. the conjugate of Fk and setting it to zero

F?k = GH

k HHk B−1

k , (5.49)

with

Bk = HkGGHHHk + σ2

k I + tr{

GGH(

Re,Txk

)TRe,Rx

k

}. (5.50)

Given the SOCP form in (5.48), the first subproblem is solved by the efficientinterior-point algorithm using the optimization toolboxes, e.g., CVX [24]. The sec-ond subproblem has the analytical solution. Thus, Algorithm 2 solves the problemby iteratively optimizing the two subproblems until the MSE targets of two succes-sive iterations is smaller than a pre-defined precision requirement ξ. This algorithmconverges to the local optimum. The main computation effect is consumed in solvingSOCP problem which is polynomial in the desired accuracy and the problem size [83],e.g., the number of users and antennas. Note that the complexity of solving the SOCPproblem is lower than the SDP problem. For the number of elementary arithmeticoperations, its upper bound could be calculated according to [13, Section 6.6.2].

5.2.2 Downlink-Based Dual-Loop Algorithm

The computational complexity of the aforementioned SOCP-based algorithm is stillrelatively high. Alternatively, the MSE minimization problem with a single sumpower constraint was solved with a more efficient constrained gradient projectionmethod [15, Section 3.3] in [57]. Motivated by this, we aim at apply this method to theconsidered problem and propose a low complexity robust solution named downlink-based dual-loop algorithm (DL-DA) in this subsection.

In order to make use of the efficient constrained-gradient projection method, wereformulate the primal problem (5.42) into a two-loop optimization problem wherepart of the optimization problem has the similar structure as in [57]. Specially, werewrite the problem (5.42) as

minG,F1,...,FK

K∑k=1

EHk {MSEk} (5.51)


s.t. tr{

A0GGH}≤ P0

tr{

ApGGH}≤ Pp, ∀ p = 1, . . . , P

where the equivalent form of interference constraint in (5.44) is used. Moreover, ac-cording to the sum-power constraint, it holds with A0 = I.

We introduce real and nonnegative auxiliary variables µi, ∀ i = 0, . . . , P, and for-malize a new problem g({µi}) by combining the multiple power constraints

g({µi}) : minG,F1,...,FK

K∑k=1

EHk{MSEk} (5.52)

s.t. tr{AGGH} ≤ P

where

A =P∑

i=0

µiAi (5.53)

P =P∑

i=0

µiPi (5.54)

The relation between the solutions of (5.51) and (5.52) is addressed in the followingpropositions.

Proposition 5.2.1. The optimal value of (5.52) provides a lower bound on that of (5.51) forarbitrary values of the auxiliary variables µi, i = 1, . . . , P.

Proof. This proof follows similarly to the derivation of [145, Proposition 4]. Specifi-cally, we can easily see that if G is a feasible solution for (5.51), it is also feasible for(5.52). Hence, the feasible region of (5.51) is a subset of that of (5.52). Consequently,the optimal value of (5.52) is equal or smaller than that of (5.51).

Proposition 5.2.2. The optimal solution of (5.51) satisfies the KKT optimality conditions of(5.52) given certain µi, ∀ i = 0, . . . , P.

Proof. We assume the optimal solution of (5.51) is represented by G? and λ?i , ∀ i =

0, . . . , P, where λi, ∀ i = 0, . . . , P are the Lagrange multipliers w.r.t. different powerconstraints, respectively. We consider the complementary slackness condition in theKKT conditions for the problem (5.51)

λi

(tr{

AiGGH}− Pi

)= 0, ∀ i = 0, . . . , P (5.55)

and the complementary slackness condition of (5.52)

λ

( P∑i=0

µi

(tr{

AiGGH}− Pi

))= 0 (5.56)


where λ is the Lagrange multiplier associated with the combined power constraint in(5.56). Similar to [145, Proposition 5], if we choose

G = G?, λ = 1, µi = λ?i , ∀ i = 0, . . . , P (5.57)

both (5.55) and (5.56) are satisfied. Similarly, follow the same derivation, the sta-tionarity, primal feasibility, and dual feasibility conditions for both problem are alsosatisfied accordingly by choosing (5.57). Hence, the optimal solution of (5.51) satisfiesthe KKT conditions of (5.52) for certain µi, ∀ i = 0, . . . , P.

In order to obtain µi, i = 0, . . . , P, we formulate the optimization problem as

max{µi}P

i=0

g({µi}) (5.58)

s.t. µi ≥ 0, i = 0, . . . , P

Based on the result in Proposition 5.2.1, the optimal value of (5.58) serves as a closelower bound to that of (5.51). Moreover, according to Proposition 5.2.2, if the KKTconditions are sufficient for (5.52), the tightness of such lower bound holds, i.e., theoptimal value of (5.51) can be attained by solving (5.58).

To summarize, two-loop optimization problem needs to be structured to obtainthe solution of (5.51): the inner loop problem (5.52) and the outer loop problem (5.58)form the complete optimization problem. In the following, we propose algorithms tosolve both subproblems.

5.2.2.1 Inner loop optimization

Considering the inner loop optimization in (5.52), it has the similar structure as thesum-MSE minimization problem with a single sum power constraint in the conven-tional multiuser network [57]. Such problem has been efficiently solved by the gradi-ent projection method. Thus, we first reformulate the considered inner-loop probleminto a standard form of the sum-MSE minimization problem with a sum power con-straint and apply this method.

Specifically, assuming L = A−1/2 and auxiliary optimization variables are intro-duced

Gk = L−HGk, G =K∑

k=1

Gk. (5.59)

The next task it to reduce the optimization variable to the equivalent precoder G.We incorporate (5.59) and (5.49) into the MSE measure (5.43), then take this newlyreformulated objective function into (5.52). It yields

g({µi}) : minG

K∑k=1

tr{INk − GHk LHH

k B−1k HkLHGk} (5.60)


s.t. tr{GGH} ≤ P

with

Bk = HkLGGHLHHHk + σ2

k I + tr{

LHGGHL(

Re,TxHk

)T}

Re,RxHk

. (5.61)

We denote the kth precoder in the lth outer loop and the nth inner loop as G(l,n)k ,

∀ k = 1, . . . , K. Inheriting the idea of the scaled constrained projection algorithmin [15, Section 3.3], G(l,n)

k is optimized iteratively via

G(l,n+1)k = Proj

[G(l,n)

k − γ(n)β(n)∆G(l,n)k

](5.62)

with γ as the step size and β is the preconditioning scalar chosen to accelerate theconvergence speed. Herein we set

β =

√√√√ P∑Kk=1

∥∥∥∆Gk

∥∥∥2 (5.63)

The update variable ∆Gk is

∆Gk := ∇∗kK∑

k=1

EHk {MSEk} (5.64)

where ∇∗k denotes the generation of the Jacobian matrix w.r.t the conjugate of matrixGk. In Appendix A.15, ∆Gk is computed as

∆Gk =− LHHk B−1

k HkLHGk +K∑

i=1

LHHk B−1

i HiLHGiGHi LHH

i B−1i HkLHGk (5.65)

+K∑

i=1

tr{

GHi LHH

i B−1i Re,Rx

HkB−1

i HiGi

}L(

Re,TxHk

)TLHGk.

The orthogonal projection in (5.62) is indicated by proj[·]. Applying the definitionin [15, Section 3.3.1] here, the projection operator guarantees the updated precodersfulfilling the power constraint

proj[G]= arg min

G

∥∥∥G− G∥∥∥

2, (5.66)

s.t. tr{

GGH}= P

Note that the inequality power constraint in (5.60) is changed to the equality in (5.66)since it is met with equality at the optimum of (5.60). This can be proved by thecontradiction. If the power of proj

[G]

is equal to P < P, then the MSE target could


Algorithm 3 DL-DA robust design

1: Initialization: Set iteration number l = 0, maximal allowable number of iterationslmax, the desired accuracy ξ, the initial precoders G(0,0)

k , ∀ k = 1, . . . , K, and the

ellipsoid center µ(0)i , ∀ i = 0, . . . , P.

2: repeat

3: l ← l + 1, for the fixed{

µ(l−1)i

}P

i=0.

4: repeat5: n← n + 16: Update G(l,n)

k via (5.62), ∀ k = 1, . . . , K.7: until MSE target converges.8: Using ellipsoid method to update µ

(l)i , ∀ p = 0, . . . , P.

9: until {µi}Pi=0 converges within the precision ξ.

be further minimization by scaling each precoder Gk, ∀ k = 1, . . . , K with√

P/P. InAppendix A.16, we obtain the result of the projection operation

proj[G]=

√√√√√ PK∑

i=1

∥∥∥Gi

∥∥∥2

F

G.

5.2.2.2 Outer Loop Optimization

The outer-loop optimization (5.58) is not necessarily differentiable. Thus, the problemcan not be directly solved by the gradient ascent method [111]. However, the sub-gradient method applies. After some mathematical manipulations, the sub-gradient{s(j)

i } at point {µ(j)i }, ∀ i = 0, . . . , P for the jth iteration is

s(j)i = tr

{G(j)

opt

(G(j)

opt

)HAi

}− Pi (5.67)

where G(j)opt is the optimal solution of (5.52) with µi = µ

(j)i , ∀ i = 0, . . . , P. To maintain

the nonnegativity of auxiliary variables, the update of µi is selected as

µi(j+1) = max

(fupdate

({µi

(j)}P

i=0, si

(j))

, 0)

(5.68)

where fupdate(·) is certain update function of µi, ∀ i = 0, . . . , P with the sub-gradientdetermined by (5.67). Herein, we use the ellipsoid method [111].

In summary, the proposed DL-DA algorithm is outlined in Algorithm 3, which iscomposed of two-loops. First, given the fixed µi, ∀ i = 0, . . . , P, the inner loop (5.52)is solved by iteratively updating the dual uplink precoding matrices via (5.62) until


convergence. Second, the outer loop (5.58) is searching for the optimal µi via thesub-gradient based method (5.68).

5.2.3 Duality-Based Dual-Loop Algorithm

In the last subsections, the non-convex problem (5.42) was solved by either the alter-nating method or the dual-loop optimization in the downlink system. In general, thedownlink problem is usually non-convex and difficult to handle due to the couplingparameters. Alternatively, the uplink problem usually has some favorable attributes,e.g., less coupling of parameters or hidden convexity [62, 145]. In order to take ad-vantage of these properties, the downlink optimization problem can be transformedinto the dual uplink problem using uplink–downlink (UL–DL) duality. Such dualityshows the reciprocity relationship between the uplink and downlink problems.

In this subsection, we refine the previous algorithms by solving the problem viathe exploitation of the developed UL-DL MSE duality, i.e., the same MSE-based targetsare achieved in both the primal downlink and the dual uplink problem. Based on thisresult, a dual-loop algorithm is proposed for the dual uplink problem. We show thatthe proposed duality-based robust algorithm has faster convergence rate and lowercomplexity compared to the downlink-based methods.

5.2.3.1 UL–DL MSE Duality with Imperfect CSI

MSE duality under a single sum power constraint was studied either with perfectCSI [57] or imperfect CSI [18]. However, in the cognitive network, both the transmitand interference power constraints need to be considered. Therefore, we need tostudy the duality with multiple power constraints. Inheriting the idea from [145]where SINR duality was established with multiple power constraints, we establishthe MSE duality with multiple power constraints and imperfect CSI, which providesthe basis of the proposed algorithm. The MSE duality is established in two steps.First, only a single power constraint and imperfect CSI is considered. Second, theabove duality is applied to the system with multiple power constraints.

MSE Duality with a Single Power Constraint We consider the downlink and uplinknetworks depicted in Figure 5.4. Although the network structure is analogous to [145],here we consider the CSI imperfection in modeling the channel matrices. Consideringthe downlink optimization problem as

minG,F1,...,FK

K∑k=1

EHk {MSEk} (5.69)

s.t. tr{AGGH} ≤ P

where the kth user’s MSE is given in (5.4). The power limit is denoted by P. Thepower constraint is the linear function of the transmit covariance matrix GGH scaled


Figure 5.4: Network structure of the primal downlink and the dual uplink with asingle power constraint.

with certain matrix A. For example, for the sum power constraint, A = I holds; forthe interference constraint in (5.44), A is replaced by Ap in (5.45).

The dual uplink problem is modeled in the right hand side of Figure 5.4. Theprecoders and equalizers are exchanged in the downlink network. K SUs transmitsignals s1, . . . , sK to the secondary BS. The uplink channel is the conjugate transposeof the corresponding downlink channel, i.e., HH

k from the kth SU to the BS. The noisecovariance matrix for every link is A. The precoders, equalizers and MSE are denotedby Gk, Fk and MSEk, respectively. Consequently, the optimization problem is

minG,F1,...,,FK

K∑k=1

EHk

{MSEk

}(5.70)

s.t.K∑

k=1

σ2k tr{

GkGHk

}≤ P

where

EHk{MSEk} = tr

{INk − FkHH

k Gk − GHk HkFH

k + FkAFHk

+K∑

l=1

Fk

(HH

l GlGHl Hl + tr

(Re,Rx

HlGlGH

l

)(Re,Tx

Hl

)T)

FHk

}. (5.71)

The duality of the primal downlink problem (5.69) and the dual uplink problem(5.70) is summarized as follows.

Theorem 5.2.1. Using the linear relation of the UL-DL precoders and equalizers as

Gk = αkFHk , Fk = α−1

k GHk , ∀ k = 1, . . . , K (5.72)


where αk is a positive scaling variable.Both the uplink and downlink systems achieve the sameuse-wise MSE subject to the same power constraint. Hence, the primal problem (5.69) andthe dual uplink problem (5.70) achieve the same MSE region with imperfect CSI by using thesame set of the precoders and the equalizers.

Proof. MSE duality indicates the following UL-DL relationship: for any set of Gk andFk of the dual uplink that achieves certain user-wise MSE, there exists at least one setof Gk and Fk of the corresponding downlink system that achieves the same use-wiseMSEs under the same sum-power consumption. Considering the linear relation of thedownlink and uplink precoders and equalizers in (5.72), in order to keep the sameuser-wise MSEs in the UL-DL conversion, the conditions EHk{MSEk} = EHk{MSEk},∀ k = 1, . . . , K should be fulfilled. After some mathematical manipulations, αk can besolved through the following linear equations:

T[α2

1, . . . , α2K

]T=[tr{

σ21 G1GH

1

}, . . . , tr

{σ2

KGKGHK

}]T(5.73)

where T is shown as follows

[T]i,j =

tr

{Fi

(K∑

l=1, l 6=iHH

l GlGHl Hl + tr

(Re,Rx

HiGlGH

l

)(Re,Tx

Hi

)T+ A

)FH

i

}, i = j

−tr{

GHi

(HiFH

j FjHHi + tr

{FH

j Fj

(Re,Tx

Hi

)T}

Re,RxHj

)Gi

}, i 6= j

.

(5.74)

Since T is the column diagonally dominant matrix, T−1 always exists with all en-tries nonnegative and the diagonal entries strictly positive. This ensures that αk hasthe feasible solution. Adding all equations in (5.73) results in the equivalent powerconsumption in (5.69) and (5.70),

K∑l=1

σ2l tr{

GHl Gl

}=

K∑l=1

tr{α2l FlAFH

l } = tr{AGGH} (5.75)

which concludes the proof.

Aforementioned duality preserves the user-wise MSEs through the UL-DL con-version. Therefore, the duality result is applicable to a group of general optimizationproblems in which the objective function is an arbitrary linear combination of theuser-wise MSEs, e.g., minimizing the worst user-wise MSE averaged over the channelrealizations within the uncertainty region.

In the special case of the sum-MSE minimization problems (5.69) and (5.70), onlyDoF equal to one is required to preserve the same sum-MSE. Thus, αk is furthersimplified by setting them to the same value [57]. Computed with the aid of thetransmit power constraint (5.75), it gives

α1 = α2 = . . . αK =

√P∑K

l=1 tr(FlAFH

l

) . (5.76)


MSE Duality with Multiple Power Constraints The MSE-duality with multiplepower constraints is exemplified by considering the problem (5.42). As discussedbefore, it can be solved through a dual-loop optimization by introducing the auxiliaryvariables µi, ∀ i = 0, . . . , P. The inner loop (5.52) is subject to a single power constraintand the outer loop (5.58) aims at the optimization of µi, ∀ i = 0, . . . , P. Applyingthis idea to the dual uplink problem, MSE duality with multiple power constraintsis given as follows: we first combine the multiple power constrains in the downlinksystem as (5.52) and apply the derived MSE duality to it. Since the problem (5.52)is in the same form as (5.69), the dual uplink problem is shown in (5.70). The sec-ond step is to determine the optimal values for nonnegative variables µi by solving(5.58), ∀ i = 0, . . . , P. Finally, the solutions obtained from the dual uplink problemis converted to (5.42) using the linear transformations (5.72). The linear transforma-tion parameters are chosen as the solutions to (5.73). In the special case of sum-MSEminimization problem in (5.42), they are in the form of (5.76).

We remark that, the MSE duality result is the extension of the duality in [57] tothe case of imperfect CSI and the duality in [18] to the case of a general form of powerconstraint as an arbitrary linear function of transmit covariance matrices. Applyingsuch duality to the primal optimization problem (5.42) yields the duality-based dual-loop optimization algorithm: the inner-loop in (5.69) and the outer-loop (5.70). Inwhat follows, we propose the efficient method to solve them. Similar to the DL-DAalgorithm in Algorithm 3, the dual uplink problem can also be solved by the efficientgradient projection method [15, 57]. The resulting algorithm is named duality-baseddual-loop algorithm (DB-DA).

5.2.3.2 Inner-Loop Optimization

We introduce an auxiliary optimization variables Gk as

Gk = σkGk. (5.77)

The weighted sum of individual power constraint is then rewritten in the form ofa standard sum-power constraint. Moreover, given the optimal fixed equalizers, theMSE of the kth user given the optimal precoders is the function of the precoders as

EHk{MSEk} = INk − σ−2k GH

k HkB−1HHk Gk (5.78)

where we use A =∑P

i=0 µiAi and

B =K∑

k=1

σ−2k

(HH

k GkGHk Hk + tr

(Re,Rx

HkGkGH

k

)(Re,Tx

Hk

)T)+ A.

The optimization problem (5.70) is then reformulated as

min{Gk}K

k=1

K∑k=1

EHk{MSEk} (5.79)


s.t.K∑

k=1

tr(

GkGHk

)≤ P.

Applying the gradient projection method [15] in (5.79), the kth precoder in the lthouter loop and the nth inner loop is updated via

G(l,n+1)k = proj

[G(l,n)

k − γ(n)β(n)∆G(l,n)k

](5.80)

where γ is the step size. Analogous to the DL-DA, β is the preconditioning scalarchosen as

β =

√√√√ P∑Kk=1∥∥∆Gk

∥∥2 (5.81)

for the sake of accelerating the convergence speed [57]. The variable ∆Gk is

∆Gk = ∇∗kK∑

k=1

EHk{MSEk}

= − σ−2k

(HkB−1HH

k Gk −K∑

l=1

(HkB−1HH

l GlGHl HlB−1HH

k Gk

)

−K∑

l=1

(tr(

σ−2l GH

l HlB−1(

Re,TxHk

)TB−1HH

l Gl

))Re,Rx

HkGk

)and the projection operator is the simple solution by scaling all precoders with a factor√

P/∑K

k=1 ‖Gk‖2F. Finally, the uplink precoders Gk, ∀ k = 1, . . . , K is obtained through

the linear relation (5.77).

5.2.3.3 Outer Loop Optimization

Considering the outer-loop optimization (5.70), the problem can be similarly solvedby a sub-gradient based method as (5.68). Herein, we also use the ellipsoid methodto update µi, ∀ i = 0, . . . , P. Combined this with the inner loop optimization, thepseudo-code of the DB-DA is summarized in Algorithm 4.

5.2.4 Performance Comparison

First, the optimality and the complexity issues of the three proposed algorithms, theSOCP-based algorithm, the DL-DA, and the DB-DA, are briefly discussed as follows.

Optimality: As addressed before, the SOCP-based algorithm is guaranteed to con-verge to local optimum. Thus, we focus on the discussion of the latter two algorithms.First we consider the inner-loop optimization problems (5.52) and (5.69) in the DL-DA and the DB-DA, respectively. They are basically applying the constrained gradi-ent projection method. Thus, the conditions under which the algorithm converges to


Algorithm 4 DB-DA robust design

1: Initialization: Set iteration number l = 0, maximal allowable number of iterationslmax, the desired accuracy ξ, the initial precoders G(0,0)

k , ∀ k = 1, . . . , K, and the

ellipsoid center µ(0)i , ∀ i = 0, . . . , P.

2: repeat

3: l ← l + 1, for the fixed{

µ(l−1)i

}P

i=0.

4: repeat5: n← n + 16: Update G(l,n)

k via (5.80), ∀ k = 1, . . . , K.7: until MSE target converges.8: Using ellipsoid method to update µ

(l)i , ∀ p = 0, . . . , P.

9: until {µi}Pi=0 converges within the precision ξ.

10: Uplink to downlink conversion using (5.72).

the solution satisfying the first-order KKT optimality conditions has been addressedin [57, Theorem 1] and [15, Chapter 3]. If the optimization problem is convex, thesolution is the global-optimum. Due to the favorable structure of the uplink problem,we find that the problem (5.69) is convex in the special case that all SUs have iden-tical Re,Tx

Hkand Re,Rx

Hk= ILK , and they transmit as many data streams as the transmit

antennas Nk = Lk, ∀ k = 1, . . . , K. Under such circumstances, the global optimum isachieved by the inner loop optimization in the DB-DA. Second, for the outer loop op-timization, we use the diminishing step size rule for the sub-gradient-based methodwhich guarantees the convergence to the optimal value [112].

Complexity: As observed in [57], the gradient projection algorithm exhibits a fastconvergence rate when applied to the sum-MSE minimization problem, whereas theSOCP-based alternating optimization converges comparably slowly especially at highSNR values. Thus, we concentrate the discussion on the gradient projection method.The main computational complexity for each of them lies in the iterative update op-erations (5.62) and (5.80). After some simple analysis, both algorithms are shown tohave the same computational order of the per-iteration operation. Therefore, the com-plexity depends on the number of iterations. In the numerical results, we comparethe number of iterations of both algorithms for some given accuracy requirements.It is shown that the proposed approach requires less iterations, which validates theadvantage of exploiting UL-DL duality.

Next, we evaluate the performance via simulation. Considering a cognitive net-work consisting of one BS and two SUs which share the same spectrum resource withtwo PUs, i.e., K = 2, P = 2, the simulation parameters are given as follows, exceptspecified in each figure. The numbers of antennas at the BS, the SUs and the PUsare Nt = 6, Lk = 2 and Lp = 1, respectively. Two independent data streams aretransmitted to a single SU, i.e., Nk = 2. Assuming the MMSE channel estimationmethod in [95] is performed, the covariance matrices of the channel error matrices areRe,Rx

Hk= σ2

esRRxHk

, Re,RxYp

= σ2epRRx

Ypand Re,Tx

Hk= Re,Tx

Yp= RTx, where σ2

es and σ2ep are the


0 3 · 10−2 7 · 10−2 0.11 0.150

0.1

0.2

Estimated error variance σ2e

Inte

rfer

ence

Pow

er

DB-DANon-robust

Figure 5.5: Average interference power vs. estimated error variance σ2e . For both SU

links, SNR=10 dB.

6 10 14 18 22 26 3010−2

10−1

100

SNR [dB]

Ave

rage

Sum

MSE

Perfect CSINon-robustSOCPDL-DADB-DA

Figure 5.6: Average sum-MSE vs. SNR of SU links.

estimated error variances of the BS-SU links and the BS-PU links, respectively. Theerror variance is represented as σ2

e and σ2e = σ2

es = σ2ep = 0.03. The transmit covariance

matrix RTx is modeled by a Toeplitz matrix with[RTx]

i,j = 0.1|i−j|. Similarly, the


Table 5.1: Average number of iterations vs. the desired accuracy ξ.

Accuracy requirement ξ 10−12 10−10 10−8 10−6 10−4

DL-DA 250 223 172 81 27

DB-DA 24 18 12 8 4

entries[RRx

Hk

]i,j

and[RRx

Yp

]i,j

are denoted by 0.2|i−j|. The accuracy of the algorithm is

given by ξ = 10−8. The transmit and interference power constraints are P0 = 0 dB andPp = −10 dB, ∀ p = 1, . . . , P.

We compare the following approaches.

• Perfect CSI: the solution is based on the assumption that the available CSI at thesecondary BS is accurate.

• Non-robust: the solution is based on the estimated CSI ignoring its imperfection.

• SOCP: proposed SOCP-based robust design in Algorithm 2.

• DL-DA : proposed downlink-based robust design in Algorithm 3.

• DB-DA: proposed duality-based robust design in Algorithm 4.

In the design of cognitive systems, the most important issue is to guarantee thatperformance degradation to the PUs is limited. Figure 5.5 plots the values of averageinterference power under the error variance σ2

e of channel estimation. The interferenceconstraint is shown to be always violated by the non-robust design while strictlyfulfilled by the robust solution, exemplified by the DB-DA. The violation gap increaseswith the larger variance due to the corresponding larger channel uncertainties.

Consider the MSE performance of the cognitive networks, Figure 5.6 demonstratesthe average sum-MSE vs. SNR values. Both robust designs outperform the non-robust solution especially in the high SNR region. The performance gap seems small.The reason is that the a primary target of robust optimization is to guarantee thenon-violation of the interference power constraint. Among the robust designs, theDB-DA achieves a marginal gain over the DL-DA, since the dual uplink problem hassome favorable properties such as hidden convexity and less coupling of parameters,which yields the better convergence behavior. Both DB-DA and the DL-DA algorithmsoutperform the SOCP-based algorithm.

From the perspective of the convergence speed, we compare the average numberof iterations of the DB-DA and the DL-DA method in Table 5.1. It is shown that theDB-DA requires less number of iterations for given desired accuracies, i.e., has fasterconvergence speed. As analyzed before, the complexity per-iteration of both methodsare of the same computational order. Therefore, the DB-DA has lower complexity.

5.3. Summary 107

5.3 Summary

Exploiting multiple antennas for the linear transceiver optimization is considered asan effective way to mitigate the interference in the underlay paradigm of a cogni-tive system. However, the performance can be deteriorated by imperfect CSI since thenon-robust design neglecting the channel uncertainties leads to the violation of the in-terference constraint and thus limits the application of cognitive systems. Two kindsof CSI error models are therefore commonly assumed: the bounded and the stochasticmodel. In this chapter, we have studied the robust transceiver optimization consid-ering both models. Specifically, on the one hand, assuming the CSI error is boundedwithin an ellipsoidal region, we employed the worst-case principle to minimize themaximum per-user MSE for any channel realization within the CSI uncertainty re-gion. An alternating algorithm was used to optimize the precoding and equalizerfilters iteratively until convergence to a local optimum is achieved. Each subprob-lem was reformulated into an equivalent convex SDP form. On the other hand, ifthe CSI error follows the stochastic model, the optimization aims at minimizing thesum-MSE of the secondary network averaged over all channel realizations within theuncertainty region. Three robust algorithms were proposed: The first one applies thealternating principle to optimize the precoding and equalizer filters iteratively. Eachsubproblem was converted into the convex SOCP form. The second one, the DL-DAalgorithm, utilized the gradient projection method [15] in the downlink transmission.The third one, the DB-DA algorithm, performed the optimization in the dual uplinkbased on the established MSE duality incorporating multiple power constraints andCSI imperfection. We also addressed the complexity and optimality issues of the pro-posed methods. The effectiveness and convergence of the algorithms were validatedand compared by numerical results. The advantage of exploiting the UL–DL dualityfor the downlink optimization is verified, in terms of both complexity and optimality.

Chapter 6

Sensing-Based Power Allocation

Up to now, we have investigated the optimization of spectrum sensing and transceiverstrategies in the interweave and the underlay paradigm, respectively. Recent workshave revealed that a significant improvement in the spectrum efficiency is obtainedby joint optimization of sensing, access, and transmission strategies [8, 21, 22, 70, 91,97, 100–102, 107, 117, 119]. This can be considered as a hybrid paradigm, i.e., a com-bination of the above-mentioned two paradigms.1 In this chapter, we apply this ideato the system considered in Chapter 4 by additionally allowing the SU to considerthe reliability of the sensing outcome and adjust its transmission strategy accordingly.Both soft-decision and hard-decision sensing results are considered. The purpose isto characterize the gain achieved by the SUs through the effective utilization of thesensing information.

In practice, the SUs only have partial CSI related to the primary link. Since suchCSI may highly depend on the location information [61], the effect of location un-certainty [125] is also considered in modeling the interference caused by the SUs tothe PUs. This idea shares some similarity with [130] where the primary exclusiveregion is determined based on the interference generated by the SUs considering therandomness of the users’ deployment. However, we are not aiming at characterizingsuch spatial opportunities for the SUs, instead we want to assess the interference levelresulting from the secondary transmission.

In what follows, we design power allocation strategies in the hybrid paradigm.After introducing the system model in Section 6.1, the cases that the sensing metricis in a general form or a special form of a binary hard decision are investigated inSection 6.2 and Section 6.3, respectively. Specific attention is paid to the hard-decisionsensing based scheme because it can be directly compared with the interweave andunderlay paradigms. In addition, we consider the location uncertainty of the primarynetwork and provide a model of interference caused by the secondary transmission inSection 6.4. The result is then incorporated into the optimization problem to designthe transmission strategy.

The results presented in this chapter are addressed in part by the author in [51]for possible future publication2.

1 In some works, e.g., [152], the term overlay corresponds to what we term interweave in this work.2 In reference to IEEE copyrighted material which is used with permission in this thesis, the IEEE

does not endorse any of RWTH Aachen University’s products or services. Internal or personal useof this material is permitted. If interested in reprinting/republishing IEEE copyrighted material foradvertising or promotional purposes or for creating new collective works for resale or redistribution,please go to http://www.ieee.org/publications_standards/publications/rights/rights_link.html to learn how to obtain a License from RightsLink.

109



110 Chapter 6. Sensing-Based Power Allocation

Figure 6.1: System model

6.1 System Model

The considered system is depicted in Figure 6.1. Compared to Figure 4.1, we addition-ally introduce the sensing link from the PT to the ST with channel power gain denotedby g0[k] at the kth time instant. The remaining notations and the input-output relationare described in Section 4.1.

Each secondary transmission process is divided into two phases: the spectrumsensing phase and data transmission phase with the duration given by Tse and Ttran,respectively. The total duration follows as

Tall = Tse + Ttran.

In the first phase, the ST calculates the sensing metric γ based on the observationsreceived from the PT through the sensing link. In the second phase, the secondarytransmit power is adapted according to γ and the available CSI. Herein, we do notrestrict the discussion to any specific sensing algorithm. Therefore, γ stands for anysensing metric such as the signal energy, sampled correlation matrix of the receivedsignals, or a binary hard decision.

Due to limited cooperation between the PUs and the SUs, the instantaneous CSIrelated to the primary links is difficult to obtain at the ST. Hence, we make the sameCSI assumptions as in Section 4.2.2. For the remainder of the chapter, we omit thetime argument for simplicity. Specifically, statistical channel parameters l12 and l21of the PT-SR and the ST-PR link are available at the ST. Additionally, the ST has theknowledge of instantaneous CSI g22. The SR is assumed to know instantaneous CSIg12 and g22.

6.2 Soft-Decision Sensing

During the sensing phase, the ST gathers the observation received from the PT andcalculates the sensing measures γ. Moreover, the ST knows the probability of the

6.2. Soft-Decision Sensing 111

presence and absence of the primary transmission, named as “ON” and “OFF” sta-tuses, i.e., p(H1) and p(H0), respectively. If γ is a continuous variable, the PDF ofγ conditioned on the status “ON” and “OFF” is given as f (γ|H1) and f (γ|H0), re-spectively; If γ is a discrete variable, we use the same notation f (γ|Hi) to denote theprobability mass function (PMF) of γ conditioned on Hi, ∀ i = 1, 2. These distributionparameters f (γ|Hi), ∀ i = 1, 2, are also known at the ST.

Consequently, given a certain γ, the probability that the primary transmission ispresent and absent can be calculated using the Bayes’ theorem as

p(H0|γ) =p(H0) f (γ|H0)

f (γ)(6.1)

p(H1|γ) = 1− p(H0|γ) (6.2)

respectively, where f (γ) is the PDF of γ with

f (γ) = p(H0) f (γ|H0) + p(H1) f (γ|H1) .

Here, we only consider the scenario in which f (γ) is non-zero, i.e., f (γ) > 0.Two power constraints are imposed. First, we limit the average interference power

received at the PR by PI if the primary transmission is active. Second, the transmitpower of the ST is restricted to PS. Both PI and PS are positive and finite. We aim atoptimizing the achievable rate of the secondary link based on the sensing metric γ andthe available CSI. Thus, introducing the optimization variable P2(γ, g22) as the powerlevel adapted to the sensing measure γ and instantaneous CSI g22, the optimizationproblem is formulated as

maxP2(γ,g22)

Eγ,g22

{p(H0|γ) ln

(1 +

P2(γ, g22)g22

σ2s

)}+ Eγ,g12,g22

{p(H1|γ) ln

(1 +

P2(γ, g22)g22

P1g12 + σ2s

)}(6.3)

s.t. P2(γ, g22) ≤ PS, ∀g22 ≥ 0P2(γ, g22) ≥ 0, ∀g22 ≥ 0Eγ,g22, g21 {P2(γ, g22)g21|H1} ≤ PI .

Using the mutual independence among the CSI of different links, e.g., g22 and g21, thelast constraint is converted into

Eγ,g22 {P2(γ, g22)|H1} ≤ PI l21 (6.4)

which has the similar form as an average transmit power constraint.

6.2.1 Optimal Power Allocation

Because the expectation and summation operations preserve convexity [115], the ob-jective function in optimization problem (6.3) is a continuously differentiable concave


function in nonnegative variable P2(γ, g22). Moreover, all the constraints are continu-ously differentiable and affine functions of P2(γ, g22). Hence, the optimization prob-lem (6.3) is a convex optimization problem. A regularity condition, e.g., the Slatercondition, is also satisfied. Consequently, the KKT conditions are the necessary andsufficient optimality conditions. In the following, we apply the KKT conditions infunction space to this optimization problem and derive the optimal solution.

We first express the objective function in (6.3) as an explicit function w.r.t. theavailable CSI. Similar to (4.32), the objective function is averaged over different chan-nel realizations of g12, it yield

Eg12, g22

{ln(

1 +P2(γ, g22)g22

σ2s + P1g12

)}= Eg22

{ln(

1 +P2(γ, g22)g22

σ2s

)+ es1 E1(s1)

}− es0 E1(s0) (6.5)

where the terms s0 and s1 are introduced for the sake of brevity

s0 =l12σ2

sP1

s1 =l12(σ2

s + P2(γ, g22)g22)

P1.

Then, the Lagrangian of (6.3) is

L = Eγ,g22

{p(H0|γ) ln

(1 +

P2(γ, g22)g22

σ2s

)}+ Eγ,g22

{p(H1|γ)

(ln(

1 +P2(γ, g22)g22

σ2s

)+ es1 E1(s1)− es0 E1(s0)

)}− λ2

(Eγ,g22 {P2 (γ, g22) |H1} − PI l21

)(6.6)

where λ2 is the non-negative Lagrangian multiplier corresponding to the constraint(6.4). Applying the KKT conditions for functional optimization [87] to (6.3), the opti-mal solution P?

2 (γ, g22) satisfies

dldP2(γ, g22)

∣∣∣∣∣P2(γ,g22)=P?

2 (γ,g22)

≥ 0, P?

2 (γ, g22) = PS

≤ 0, P?2 (γ, g22) = 0

= 0, 0 < P?2 (γ, g22) < PS

(6.7)

λ?2 ≥ 0 (6.8)

λ?2(Eγ,g22 {P2(γ, g22)|H1} − PI l21

)= 0 (6.9)

Eγ,g22 {P2(γ, g22)|H1} ≤ PI l21 (6.10)

6.2. Soft-Decision Sensing 113


l = p(H0|γ) ln(

1 +P2(γ, g22)g22

σ2s

)+ p(H1|γ)

(ln(

1 +P2(γ, g22)g22

σ2s

)+ es1 E1(s1)− es0 E1(s0)

)− λ2

(p (H1|γ)

p(H1)P2(γ, g22)− PI l21

). (6.11)

and the derivative on the left hand side of (6.7) is

dldP2(γ, g22)

= (p(H0|γ) + p(H1|γ))1

σ2s /g22 + P2(γ, g22)

+ p(H1|γ)(

l12g22

P1es1 E1(s1)−

1σ2

s /g22 + P2(γ, g22)

)− λ?

2p (H1|γ)

p(H1). (6.12)

We multiply both sides of (6.12) with positive f (γ) and insert (6.1) and (6.2) intoit. After some mathematical manipulations, we obtain P?

2 (γ, g22) from solving KKTconditions

P?2 (γ, g22) =

0, H2(α0,h, α1,h, 0) ≤ λ?

2 f (γ|H1)

p?(γ, g22), H2(α0,h, α1,h, PS) < λ?2 f (γ|H1) < H2(α0,h, α1,h, 0)

PS, H2(α0,h, α1,h, PS) ≥ λ?2 f (γ|H1).

(6.13)

with H2(α0, α1, x) given as

H2(α0, α1, x) =α0P1

l12 (σ2s + xg22)

+ α1el12(σ2

s +xg22)P1 E1

(l12(σ2

s + xg22)

P1

)(6.14)

and

α0,h = p(H0) f (γ|H0)l12g22

P1

α1,h = p(H1) f (γ|H1)l12g22

P1.

In (6.13), the value p?(γ, g22) is equal to the solution x of the following equation

H2(α0,h, α1,h, x) = λ?2 f (γ|H1). (6.15)

Using the result in Appendix A.4, it is easy to verify that H2(α0,h, α1,h, x) is strictlymonotonically decreasing on x, x ∈ (0, PS), for finite and positive g22. Therefore, thebisection method can be applied to search for the root of (6.15).


The variable λ?2 is determined to satisfy the constraint (6.4), similar in the super-

vised thesis [104]. More particularly, If PS ≤ PI l21, the ST can always transmit withpower PS while the average interference power constraint is strictly satisfied. Conse-quently, the computation of λ?

2 is not required. If PS > PI l21, the optimal λ?2 should be

chosen to let the inequality (6.4) be satisfied with equality. We remark that λ∗2 is finiteand positive in this case. Specifically, first, if λ∗2 = 0 holds, then the optimal solution(6.13) results P∗2 (γ, g22) = PS for any positive g22 by noting that H2(α0,h, α1,h, x) in(6.14) has positive value for non-negative x and finite g22. This violates the averagepower contraint since the average power under such circumstance is equal to PS andlarger than PI l21. Second, if λ∗2 is infinitely large, this yields P∗2 (γ, g22) = 0 which isobviously not optimal.

Based on the above discussion, the optimal solution for the case PS ≤ PI l21 isP∗2 (γ, g22) = PS, which can be straightforwardly obtained. Therefore, we focus on thesolution for the case PS > PI l21 in which λ∗2 is finite and positive.

6.2.2 Suboptimal Power Allocation

Revising the optimal strategy, the relationship between the solution (6.13) and theeffect of the CSI is not explicitly shown. Moreover, a real-time numerical calculationof p?(γ, g22) is required for each g22 within its feasible region. Thus, the optimalstrategy is time-consuming. In this subsection, we propose a low complexity subop-timal power allocation. The relation between the power level and the CSI is explicitlydescribed.

Motivated by the effectiveness of the suboptimal strategy in Section 4.2.2.2, we usethe following approximation to simplify the optimal solution

exE1(x) ≈ 1x + 1

. (6.16)

Specifically, the first condition in (6.13), H2(α0,h, α1,h, 0) ≤ λ?2 p(γ|H1), is approximated

by

g22 ≤λ?

2 f (γ|H1)

p(H0) f (γ|H0)/σ2s + p(H1) f (γ|H1)/(σ2

s + P1/l12)(6.17)

which shows that the optimal strategy is to switch off the transmission when g22 issmaller than a threshold glow with

glow =λ?

2 f (γ|H1)

p(H0) f (γ|H0)/σ2s + p(H1) f (γ|H1)/(σ2

s + P1/l12). (6.18)

Similarly, the condition H2(α0,h, α1,h, PS) ≥ λ?2 f (γ|H1) is approximated by

p(H1) f (γ|H1)

(σ2s + P1/l12)/g22 + PS

+p(H0) f (γ|H0)

σ2s /g22 + PS

≥ λ?2 f (γ|H1) (6.19)

in which the left hand side is a monotonically increasing function of g22. Hence, thereexists a gupp such that g22 ≥ gupp indicates the satisfaction of the condition (6.19). The

6.3. Hard-Decision Sensing 115

solution of gupp is calculated in Appendix A.17. Applying the approximation (6.16)to (6.15), we obtain an approximation of p?(γ, g22) as

p?(γ, g22) =xP1/l12 − σ2

sg22

(6.20)

where x is calculated by a similar method as discussed in Appendix A.17:

x =α0,h + α1,h − λ?

2 f (γ|H1) +√(

α0,h + α1,h − λ?2 f (γ|H1)

)2+ 4α0,hλ?

2 f (γ|H1)

2λ?2 f (γ|H1)

.

(6.21)In summary, the suboptimal power control strategy is

P?2 (γ, g22) =

0, g22 ≤ glow

p?(γ, g22), glow < g22 < gupp

PS, otherwise.

(6.22)

6.3 Hard-Decision Sensing

Hard-decision sensing-based power control is encompassed as a special case of soft-decision sensing-based scheme considered in Section 6.2. Specifically, the sensingmeasure γ is reduced to binary statuses H0 and H1 representing the declaration ofthe primary transmission as present or absent, respectively. However, a specific inves-tigation on this problem is still informative since it can be directly compared to thestandard underlay and interweave paradigms. Thus, the result of this problem servesthe purpose to exemplify the performance gain over the reference paradigms. In thissection, we address the optimal and suboptimal power allocation solutions based onhard-decision sensing. Note that the notation f (·) indicates a PMF. Due to the closerelation to Section 6.2, we omit the detailed derivations and present the final results.

6.3.1 Optimal and Suboptimal Power Allocation

Simplifying the sensing metric γ into two statuses H0 and H1, we have

γ = H0 ⇒{

f (γ|H0) = 1− PFA

f (γ|H1) = 1− PD(6.23)

γ = H1 ⇒{

f (γ|H0) = PFA

f (γ|H1) = PD(6.24)

Inserting them into the optimization problem (6.3), it is simplified to

maxP2,0(g22), P2,1(g22)

Rhd (6.25)


s.t. P2,0(g22) ≤ PS, P2,1(g22) ≤ PS ∀g22 ≥ 0P2,0(g22) ≥ 0, P2,1(g22) ≥ 0, ∀g22 ≥ 0Eg22 {(1− PD)P2,0(g22) + PDP2,1(g22)} ≤ PI l21

where the objective function is

Rhd = Eg22

{p(H0)

((1− PFA) ln

(1 +

P2,0(g22)g22

σ2s

)+ PFA ln

(1 +

P2,1(g22)g22

σ2s

))}+ Eg12,g22

{p(H1)

((1− PD) ln

(1 +

P2,0(g22)g22

P1g12 + σ2s

)

+ PD ln(

1 +P2,1(g22)g22

P1g12 + σ2s

))}. (6.26)

Since the problem (6.25) is the special case of the problem (6.3) based on soft decisions,the optimal solutions are derived based on (6.13) by incorporating the PMFs (6.23) and(6.24). We directly present the results as follows:

P?2,0(g22) =

0, H2(α00,hd, α10,hd, 0) ≤ λ?

2(1− PD)

p?0(g22), H2(α00,hd, α10,hd, PS) < λ?2(1− PD) < H2(α00,hd, α10,hd, 0)

PS, otherwise(6.27)

P?2,1(g22) =

0, H2(α01,hd, α11,hd, 0) ≤ λ?

2PD

p?1(g22), H2(α01,hd, α11,hd, PS) < λ?2PD < H2(α01,hd, α11,hd, 0)

PS, otherwise.

(6.28)

with

α00,hd = p(H0)(1− PFA)l12g22

P1

α10,hd = p(H1)(1− PD)l12g22

P1.

α01,hd = p(H0)PFAl12g22

P1

α11,hd = p(H1)PDl12g22

P1.

The notations P?2,0(g22) and P?

2,1(g22) represent the optimal power control under thesensing decision H0 and H1, respectively. The intermediate power values p?0(g22) andp?1(g22) can obtained as the roots of the following equations

H2 (α00,hd, α10,hd, p?0(g22)) = λ?2(1− PD) (6.29)

H2 (α01,hd, α11,hd, p?1(g22)) = λ?2PD (6.30)


respectively, through some numerical method such as the bisection method.The efficient suboptimal strategy can be derived analogously to (6.22) by incorpo-

rating the PMFs (6.23) and (6.24).

6.3.2 Referenced Strategies

In order to provide a thorough comparison between the different standard cognitiveradio paradigms, the development of power allocation strategies is required for thestandard paradigms. In this subsection, we show that the power optimization problemin both interweave and underlay paradigm can be formulated as a special case ofthe sensing-based power control in Section 6.3.1. Hence, the methodology for thederivation of the optimal and suboptimal power allocation strategies in Section 6.3.1can be straightforwardly applied.

6.3.2.1 Interweave Paradigm

In the interweave paradigm the SUs opportunistically access the spectrum only if theydeclare the inactivity of the primary transmission. Thus, the optimization problem isformulated as

maxPin(g22)

Rin (6.31)

s.t. Pin(g22) ≤ PS, ∀g22 ≥ 0Pin(g22) ≥ 0, ∀g22 ≥ 0(1− PD)Eg22 {Pin(g22)} ≤ PI l21

with

Rin = Eg12,g22

{p(H0)(1− PFA) ln

(1 +

Pin(g22)g22

σ2s

)

+ p(H1)(1− PD) ln(

1 +Pin(g22)g22

P1g12 + σ2s

)}(6.32)

which is a special case of (6.26) by setting P2,0(g22) = Pin(g22) and P2,1(g22) = 0,∀ g22 ≥ 0, i.e., the ST only transmits under the decision H0. The power allocationstrategies to solve (6.33) can be straightforwardly derived from Section 6.3.1.

6.3.2.2 Underlay Paradigm

In the underlay paradigm, the power allocation strategy is designed assuming theprimary transmission is always active. Consequently, the optimization problem is

maxPun(g22)

Run (6.33)

s.t. Pun(g22) ≤ PS, ∀g22 ≥ 0


Pun(g22) ≥ 0, ∀g22 ≥ 0Eg22 {Pun(g22)} ≤ PI l21

with

Run = Eg12,g22

{p(H0) ln

(1 +

Pun(g22)g22

σ2s

)+ p(H1) ln

(1 +

Pun(g22)g22

P1g12 + σ2s

)}. (6.34)

We note that (6.34) is a special case of (6.26) by setting P2,0(g22) = P2,1(g22), ∀ g22 ≥ 0,i.e., the power level is not adapted to the sensing observations. The power allocationstrategies to solve (6.33) can be directly obtained from the results in Section 6.3.1.

For the evaluation of the proposed strategies, we set the simulation parametersas follows. The statistical CSI parameter is l22 = 1, l21 = 1, and l12 = 2. The noisevariance of the secondary link is σ2

s = 1. All power values in dB are given relative tothe reference level of value 1. The transmit power of the PT is 10 dB. The interferencepower limit PI is chosen between three values −5, 0, and 5 dB. The probabilities thatthe primary transmission is active and inactive are equal, i.e., p(H0) = p(H1) = 0.5.According to the requirements of IEEE 802.22 WRAN, the performance of spectrumsensing should guarantee the false alarm rate below 10% and a probability of detec-tion above 90%. Herein, the ST uses energy detection as the sensing algorithm withthe observation length equal to 30. We choose the threshold on the ROC curve ofenergy detection that it yields PFA = 9.84% and PD = 92.32%.

Power allocation strategies for different paradigms are compared. They are listedas follows.

• Optimal algorithm for hybrid paradigm based on hard-sensing decisions: thesolution is given in (6.27) and (6.28).

• Suboptimal algorithm for hybrid paradigm based on hard-sensing decisions: thepower value is in the form of (6.22) by incorporating (6.23) and (6.24).

• Suboptimal algorithm for the interweave paradigm: the solution is similar tothat of the hybrid paradigm as discussed in Section 6.3.2.1.

• Suboptimal algorithm for underlay paradigm: the solution is similar to that ofthe hybrid paradigm as elaborated in Section 6.3.2.2.

Note that in order to reduce the computational complexity, only the suboptimal strate-gies in the reference paradigms are considered. This idea is based on the observationthat the near-optimality of the suboptimal power allocation strategies is achieved forthe hybrid paradigm, as evidenced by the following numerical results.

Figure 6.2 shows the achievable rate of the secondary link vs. the ratio ρ = PS/PI .We only consider the case ρ > 1 since the interference power constraint is usuallystricter than the transmit power constraint. The detailed explanations of the plots aregiven below the figure. The near-optimality of the suboptimal algorithm is validatedfor the hybrid paradigm. Moreover, the power adaptation in the underlay paradigmsuffers from a great performance loss compared to the hybrid one due to the lack of


2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 320

0.5

1

1.5

2

2.5

ρ = PS/PI

Ach

ieva

ble

rate

R[b

its/

s/H

z]

PI = −5 dBPI = 0 dBPI = 5 dB

Figure 6.2: Achievable rate of the secondary link vs. the ratio ρ = PS/PI . Solidlines: optimal algorithm for the hybrid paradigm based on hard-sensingdecisions; circles: suboptimal algorithm for the hybrid paradigm basedon hard-sensing decisions; solid lines marked with crossings: suboptimalalgorithm for the interweave paradigm; solid lines marked with squares:suboptimal algorithm for the underlay paradigm.

sensing observations. Finally, the hybrid paradigm also outperforms the interweaveone due to the additional transmission during the cases even the presence of theprimary transmission is detected.

Remarks: Strictly speaking, the algorithm for the underlay paradigm is not eval-uated in a “fair” manner compared with other paradigms. Specifically, the actualtransmission process of the underlay paradigm only requires the data transmissionphase, which is in contrary with the hybrid and interweave ones which require thetwo phases protocol, i.e., spectrum sensing and data transmission phases during eachprocess. A fairer comparison is to scale its achievable performance of the underlayparadigm with the factor Tall/Ttran, where Tall and Ttran are the duration of two phasesand data transmission phase, respectively. Due to the fact that the duration of twophases is larger than a sole phase, i.e., Tall/Ttran > 1, the performance of the underlayparadigm shown here is the lower-bound of the actual performance. However, suchduration lengths are not clarified in the current standard and the study on this issueis still ongoing. Hence, in order to keep consistent with the other paradigms, weassume here the underlay paradigm also employs the two phases process but withthe sensing phase wasted for usage. If the sensing phase is short enough comparedwith the data transmission phase, i.e., Tall/Ttran ≈ 1, herein the performance of theunderlay paradigm asymptotically represent the actual performance.


Figure 6.3: System model considering the random deployment of the users.

6.4 Interference Modeling

Location uncertainty is one of the challenges in realistic CR systems. For instance,in case the primary network is a TV network, the PRs are passive devices and theirexact locations may be unknown to the SUs. On the contrary, the location of the PTcan be easily obtained. Another example is a primary downlink cellular network inwhich the position of the BS is fixed and the mobile terminals’ position information isdifficult to be acquired. Given such location uncertainty, the performance degradationcaused to the PUs is hard to control at the SUs. In this section, we study a way tocalculate the expectation value of the interference to the PR caused by the ST consid-ering the randomness of the PR’s position. This result can be employed in Section 6.2and Section 6.3 to design the power adaptation strategies.

The statistical CSI parameter can be obtained depending on the distance betweenthe users. For example, using the path-loss channel model in [61, 117] and denotingthe distance of the ST-PR link as d21, the statistical CSI l21 is

l−121 = Kd−α

21 (6.35)

where the path loss exponent α is within the typical range [1.6, 6] [109]. The variableK is a constant which incorporates the effect of transmit and receiver antennas anddepends on the frequency band [61]. Without loss of generality, we set K = 1 in theremainder of this section. Note that the shadowing effect is currently not considered.The same relation as stated in (6.35) applies to the statistical CSI and the distances ofother links, respectively.

Considering the effect of location uncertainty on the interference constraint, e.g.,the left hand side of the third constraint in (6.3), the expected interference power is

Eγ,g22,g21 {P2(γ, g22)g21|H1}

6.4. Interference Modeling 121

(a)= El21

{Eγ,g22,g21 {P2(γ, g22)g21|l21,H1}

}(b)= El21

{Eγ,g22 {P2(γ, g22)|H1}

l21

}(c)= Eγ,g22 {P2(γ, g22)|H1}Ed21

{d−α

21}

(6.36)

where, in (a), we use the definition of the conditional PDF. In (b), the mutual indepen-dence between the channel power gain g22 and g21 is applied. In (c), we incorporate(6.35) and use K = 1. Therefore, in order to model the interference from the ST tothe PR or constrain the interference power level in the optimization problem, e.g,in the optimization problem (6.3), it is imperative for the SU to learn the distanceinformation d21 and calculate the term Ed21

{d−α

21

}.

Figure 6.3 shows the system consisting of one PT-PR link and one ST.3 We assumethat the PR is randomly and uniformly distributed inside a circle of radius R1 centeredat the PT. The PDF of the distance of the PT-PR link d11 is

fd11(x) =

2xR2

1, 0 ≤ x ≤ R1

0, otherwise.(6.37)

The ST is assumed to be located outside the primary network with ε protection region[130], i.e., the minimum distance between the ST and the PR is ε. The distance ofthe PT-ST link is given by d0 and d0 > R1 + ε holds. The angle between the PT-PRlink and the PT-ST link is denoted by θ and thus θ is uniformly distributed in [0, 2π).The ST and the SR have the exact position knowledge of the PT but only know thedistribution of the PR.

Using the law of cosines, we have

d21 =√

d211 − 2d11d0 cos θ + d2

0 (6.38)

Incorporating it into the expectation Ed21

{d−α

21

}, we have

Ed21

{d−α

21}

(a)= Ed11,θ

{(d2

11 − 2d11d0 cos θ + d20

)−α/2}

(b)=

∫ R1

0

∫ 2π

0

(d2

11 − 2d11d0 cos θ + d20

)−α/2 12π

2d11

R21

dθd(d11)

=1

πR21dα

0

∫ R1

0

∫ π

02d11

((d11

d0

)2

− 2(

d11

d0

)cos θ + 1

)−α/2

dθd(d11) (6.39)

3 The SR is omitted herein since we focus on calculating the interference caused to the PR assuming itslocation ambiguity.


where, in (a), we insert (6.38). Note that since the exact knowledge of d21 is notavailable at the ST, the expectation Ed21

{d−α

21

}is calculated w.r.t. the related random

variables θ, d11, and d0. In (b), we incorporate the PDF of θ and d11. In general,the closed-form solution of the integral (6.39) is difficult to obtain. However, for thespecial case that α is a even number, i.e., α = 2i, i ∈N, it can be rewritten as

Ed21

{d−α

21}

(a)=

1πR2

1dα0

∫ R1

02d11π

(1−

(d11

d0

)2) α

2

α2−1∑k=0

(α2 + k− 1

)!

(k!)2 (α2 − k− 1

)!

(

d11d0

)2

1−(

d11d0

)2

k d(d11)

(b)=

1R2

1dα−20

∫ (R1d0

)2

0

1(1− ξ)

α2

α2−1∑k=0

a(α)kξk

(1− ξ)k dξ

=1

R21dα−2

0

α2−1∑k=0

a(α)k

∫ (R1d0

)2

0

ξk

(1− ξ)(α2+k)

dξ (6.40)

where, in (a), we apply [52, Eq. 3.616.2]. In (b), we use ξ = (d11/d0)2 and define

a(α)k =

(α2 + k− 1

)!

(k!)2 (α2 − k− 1

)!. (6.41)

A closed-form expression of (6.40) for even number α can be given as follows.

• α = 2:

Ed21

{d−α

21}=

1R2

1

α2−1∑k=0

a20

∫ (R1d0

)2

0

1(1− ξ)

dξ =1

R21

ln

(d2

0

d20 − R2

1

). (6.42)

• α = 4, 6, 8, . . .: we define

I(k; α) =

∫ (R1d0

)2

0ξk(1− ξ)−(

α2+k)dξ (6.43)

and thus Eq. (6.40) is reformulated as

Ed21

{d−α

21}=

1R2

1dα−20

α2−1∑k=0

a(α)k I(k; α). (6.44)

Using integration by parts, we find I(k; α), ∀ k > 0, can be calculated recursively

I(k; α) =ξk

1− α2 − k

(1− ξ)1− α2−k

∣∣∣∣∣(

R1d0

)2

0

6.4. Interference Modeling 123

− 11− α

2 − k

∫ (R1d0

)2

0ξk−1(1− ξ)−(

α2+k−1)dξ

=ξk

1− α2 − k

(1− ξ)1− α2−k

∣∣∣∣∣(

R1d0

)2

0

− I(k− 1; α) (6.45)

with

I(0; α) =(1− ξ)1−α

1− α2

. (6.46)

For example, for the path loss exponent α = 4, we have

Ed21

{d−α

21}=

(a(4)0 I(0; 4) + a(4)1 I(1; 4)

)R2

1d20

=1

(R21 − d2

0)2

. (6.47)

So far, we have discussed the closed-form expression of Ed21

{d−α

21

}for positive

and even numbers of α. In a realistic scenario, the path loss exponent α typicallyvaries from 1.6 to 6. Therefore, it is also desirable to consider the cases where α is anyfraction within this region. However, a closed-form expression for this general caseis difficult to derive. We resort to two alternatives. On the one hand, some existingnumerical integration methods, e.g., trapezoidal numerical integration, can be appliedto (6.39). The details are omitted here. On the other hand, we can derive an upperbound of Ed21

{d−α

21

}denoted as DU. The purpose to calculate its upper bound is

reasoned as follows. Replacing Ed21

{d−α

21

}by its upper bound DU in (6.36), we have

Eγ,g22 {P2(γ, g22)|H1}Ed21

{d−α

21}≤ Eγ,g22 {P2(γ, g22)|H1}DU ≤ PI

⇒ Eγ,g22 {P2(γ, g22)|H1} ≤ PI/DU ≤ PI/Ed21

{d−α

21}

(6.48)

where PI is the interference power limit at the PR. Eq. (6.48) indicates that the powerconstraint is bounded by the lower bound of the original power limit, thus a stricterpower constraint is imposed. Incorporating this into the optimization problem, weobtain a suboptimal power allocation solution that strictly guarantees the interferencecaused to the PUs.

In order to establish DU, we consider the system model shown in Figure 6.4. Com-pared to Figure 6.3, apart from the primary network centered at the PT with the radius

R1, we additionally consider two networks centered at the ST with radii√

d20 − R2

1 andd0− R1 which are drawn with dash-dotted and dashed lines, respectively. It is easy to

verify that√

d20 − R2

1 ≥ d0 − R1 holds under the previous assumption d0 > R1 + ε. Tocalculate the upper bound DU, we assume the PR is only uniformly distributed in theshadowed region. Since the PR in this region is in general closer to the ST comparedto the primal assumption that the PR is uniformly deployed in the circle centered atthe PT, the expected value Ed21

{d−α

21

}is upper-bounded for positive α.


Figure 6.4: System model considering the random deployment of the users. Thesolid circle denotes the primary network. The dash-dotted and dashed

circles are centered at the ST with radii√

d20 − R2

1 and d0 − R1, respec-tively. The PR is uniformly distributed in the shadowed region.

Under the aforementioned assumptions, the PDF of the distance d21 of the ST-PRlink is

fd21(x) =

2x

d20 − R2

1 − (d0 − R1)2, d0 − R1 ≤ x ≤

√d2

0 − R21

0, otherwise.(6.49)

Thus, the upper bound of Ed21

{d−α

21

}is computed as

DU =

∫ √d20−R2

1

d0−R1

d−α21 fd21(d21)d(d21)

=

2

(α− 2)(d2

0 − R21 − (d0 − R1)2

) 1(d0 − R1)α−2 −

1(√d2

0 − R21

)α−2

, α 6= 2

2(d2

0 − R21 − (d0 − R1)2

) (ln((√

d20 − R2

1

))− ln (d0 − R1)

), otherwise.

For realistic scenarios, we only need to use this analytical result for α ∈ [1.6, 6] andα is not necessarily an even number. If α is the even number, e.g., α = 2, we use thederived exact expression in (6.42).

The accuracy of interference modeling is verified in Figure 6.5. Assuming the PTis located at the origin of the cartesian coordinate system. A ST is positioned at (2, 2).

6.5. Summary 125

0.1 0.3 0.5 0.7 0.9 1.1

5

10

15·10−2

Radius of the primary network R1

Expe

cted

valu

eof

Ed 2

1

{ d−α

21

}

α = 2α = 3α = 4

Figure 6.5: Expected value of Ed21

{d−α

21

}vs. the radius of the primary network R1

under different path loss exponents. Solid lines with squares: analyticalresults (exact value for even number α and an upper bound for a generalvalue of α). Solid lines with circles: numerical results.

The primary network is a circle centered at the PT and R1 is the radius varying from0.1 to 1.1. The exemplified path loss exponent α is chosen to be 2, 3, and 4. Theresults show that the analytical result matches the numerical evaluation well for evennumbers of α. On the other hand, the upper bound of Ed21

{d−α

21

}is more accurate

when the radius R1 is smaller, i.e., the ST is far away from the primary network.Figure 6.6 shows the achievable rate of the secondary link vs. the transmit power

constraint PS. The random deployment of the PR is assumed with the path lossexponent α = 2 (free space). The detailed explanations of the plots are given below thefigure. Similar to Figure 6.2, the near-optimality of the suboptimal algorithm for thehybrid paradigm is also verified. Moreover, the superiority of the power adaptationstrategies in the hybrid paradigm is validated against the standard underlay and theinterweave paradigm.

6.5 Summary

In this chapter, we have verified the effectiveness of a hybrid paradigm, i.e., a combi-nation of the standard interweave and underlay paradigms. Specifically, we allowedthe SU to exploit the spectrum sensing information at the ST to adapt its transmissionparameters, e.g., the power level. The effect of location uncertainty is also consid-ered in modeling the interference caused to the primary system. Several observationsare obtained: On the one hand, compared to the power allocation in the interweave


0 2 4 6 8 10 12 14 16 18 20 22 240

1

2

3

4

Transmit power limit PS [dB]

Ach

ieva

ble

rate

R[b

its/

s/H

z] PI = −5 dBPI = 0 dBPI = 5 dB

Figure 6.6: Achievable rate of the secondary link vs. the transmit power limitPS with different interference power limits PI . The random deploy-ment of the PR is assumed. Solid lines: optimal algorithm for thehybrid paradigm based on hard-sensing decisions; circles: suboptimalalgorithm for hybrid paradigm based on hard-sensing decisions; solidlines marked with crossings: suboptimal algorithm for the interweaveparadigm; solid lines marked with squares: suboptimal algorithm forthe underlay paradigm.

paradigm, the performance gain achieved by the proposed strategy in the hybridparadigm is mainly due to the additional transmission during the period when thepresence of the primary user is detected. On the other hand, given the reliable spec-trum sensing results and comparably short sensing duration, the proposed schemeoutperforms the underlay paradigm due to the additional utilization of sensing ob-servations.

Overall, we have studied the joint system design of the sensing and transmis-sion functionals. During the last few decades, research focusing on some specificalfunctionals for the secondary system has been very active, e.g., on spectrum sensing,access protocol, or transceiver design. Therefore, we envision large potentials in thejoint consideration of these functionals in a CR system design to finally achieve “realcognition”.

Chapter 7

Conclusions and Outlook

Recognized as a promising technology, cognitive radio allows for the coexistence ofboth licensed and unlicensed devices. It builds on the environment-aware dynamicspectrum access aiming at increasing spectral efficiency and restraining the perfor-mance degradation to the incumbent devices. This demands for a proper interferencemanagement for secondary systems, including the identification of the PUs’ activityand adaptation of the transmission parameters accordingly. However, the design ofstrategies at the SUs is challenging due to limited cooperation between the PUs andthe SUs. Driven by this fact, the focus of this thesis is a comprehensive and realisticstudy on interference management strategies and the characterization of the achiev-able performance of the secondary link in secondary systems.

Spectrum sensing aims at detecting the presence or absence of the PUs. Suchsensing decision can be exploited for interference avoidance in opportunistic trans-mission, or the adjustment of transmission parameters in spectrum sharing. The mainchallenge encountered in spectrum sensing is the high requirement on sensitivity, re-liability, and agility, especially in the low SNR region with environment uncertainties,e.g., fading channels, noise uncertainty, and the hidden primary user problem. Co-operative spectrum sensing is considered in order to address the hidden primaryuser problem by exploiting the DoFs offered by multiple SUs. Mathematically, suchproblems subject to environment uncertainties are formulated as binary hypothesisproblems with unknown parameters which can be effectively solved by the GLRTprinciple. In Chapter 3, we applied the GLRT framework and shown how to effi-ciently utilize the limited a priori knowledge at the SUs to combat the environmentuncertainties. Specifically, on the one hand, the usage of rank information to extractthe structure of the primary signal space was elaborated. Based on the estimation ofthe signal structure, subspace-based cooperative sensing methods are proposed whichdifferentiate between binary hypotheses using the distinct correlation properties ofthe received signals. The quantitative comparison with other conventional coopera-tive sensing methods has revealed the effectiveness of using the rank information andthe robustness to the uncertainties of the signal space and noise variances. On theother hand, we considered the effect of the fading sensing and reporting channels oncooperative spectrum sensing. Due to practical challenges, the SUs only have partialCSI of the channels and do not know the exact structure of the primary signal space.We derived GLRT-based sensing algorithms under two circumstances, i.e., when thenoise variances of the sensing channels are unknown or known. The resulting meth-ods counter the performance degradation introduced by the fading effects and thenoise uncertainty.

127

128 Chapter 7. Conclusions and Outlook

Interference mitigation techniques are inevitably required in spectrum sharingsystems. For example, the concurrent operation of the PUs and the SUs occurs inthe underlay and the overlay paradigm, as well as in case of a missed detection inthe interweave paradigm. Contrary to conventional interference networks consistingof users with equal priority to access the resources, the unique challenge in DSA isthe limited preliminary knowledge related to primary systems available at the SUs.Based on this limited information, the SUs need to optimize their transceiver strate-gies, such as transmit power, bandwidth, or precoders and equalizers. The goal is toachieve the tradeoff between improved secondary network throughput and, most crit-ically, constrain the performance loss of the primary transmission. Moreover, giventhe developed strategies, it is also desirable to quantitatively characterize the achiev-able performance of the SUs at the expense of the performance degradation to thePUs. Such an analysis can be used for a performance assessment or facilitate thedecision in selecting system parameters. Hence, we addressed these issues in Chap-ter 4 and Chapter 5. In particular, for a single-antenna spectrum sharing system, westudied power allocation strategies for the SUs subject to different QoS constraintson the primary link: a traditional interference temperature constraint and an out-age probability constraint. Not only the optimal and low-complexity near-optimalpower allocation strategies are developed, but also the achievable performance of thesystem is approximately evaluated in closed form. Additionally, for sophisticatedmulti-antenna infrastructure-based spectrum sharing networks, we aimed at devel-oping the robust transceiver filters at the SUs subject to the interference temperatureconstraint under the consideration of imperfect CSI. Two commonly-assumed CSI er-ror models are used: a bounded CSI error model and a stochastic CSI error model.The difficulty in solving such problems lies in their non-convexity. We derived sev-eral efficient algorithms by applying different approaches. For example, we proposedthe alternating method in which the non-convex primal problem is decomposed andreformulated into convex subproblems, the constrained gradient projection methodwhich is in general more efficient than the methods depending on convex optimiza-tion solvers, and the method based on uplink–downlink duality in which the favorableproperty of the dual uplink problem can be exploited to solve the primal downlinkproblem. Through a performance evaluation and comparison, we verified that therobust transceiver design is able to strictly limit the performance degradation of thePUs in spite of the imperfect CSI at the SUs, which is a critical criterion for the de-sign of the secondary transmission. Furthermore, the advantage of exploiting theconstrained gradient projection method, especially by combining this method withuplink–downlink duality, is validated to solve the downlink optimization problem interms of faster convergence rate and better target value.

So far, spectrum sensing and interference mitigation strategies were investigatedonly in a single cognitive paradigm, e.g., either in the interweave paradigm or the un-derlay paradigm, respectively. In Chapter 6, we studied a hybrid paradigm combiningthe interweave and the underlay paradigm. Specifically, we extended the design ofpower allocation strategies in Chapter 4 to the scenario allowing the ST to exploit thespectrum sensing information to adapt the power level. The effect of location uncer-

129

tainties is also considered in modeling the interference caused to the primary system.Both soft-decision and hard-decision sensing results are explored. We found thatcompared to the power allocation in the interweave paradigm, the proposed strategyin the hybrid paradigm achieves better performance due to transmission upon thedetection of the PUs’ activity, whereas it outperforms the underlay paradigm due tothe additional utilization of sensing observations given reliable sensing results andcomparably short sensing durations. Therefore, we have exemplified the essential“cognitive” property that acquiring the useful information from the environment andutilizing such information improves the spectrum efficiency.

Within this thesis, we have conducted research on advanced interference manage-ment techniques in CR systems by considering realistic challenges. For the futurework, we envision the following topics.

• Optimization of spectrum sensing parameters: In the proposed sensing methods,the spectrum sensing parameters, e.g., the sensing observation length and thenumber of cooperative users, are fixed. It would be interesting to dynamicallyoptimize such parameters, e.g., to optimize the sensing and transmission lengthjointly in one frame to achieve the tradeoff between the sensing effort and theresulting throughput gain. Furthermore, due to the difficulty in deriving closed-form relations between the probability of detection and the false alarm rate,the decision threshold is in general hard to optimize. Therefore, an exact oran asymptotic performance analysis of certain sensing methods, e.g., similarto [10, 136], is imperative for the selection of the threshold.

• Extension of power control strategies to the multiuser scenario: Based on our researchon the power allocation strategies in a spectrum sharing system consisting of asingle primary and a single secondary link, we are interested in answering thequestion of how the current result scales with a larger number of users in thesecondary network. Moreover, cognitive transmission is potentially suitable forshort-range transmission due to its low power emission property [54]. Therefore,the extension of the present work to such short-range transmission is also ofinterest, including how multiple SUs with different QoS requirements exploitthe dynamic power allocation to share the available spectrum opportunities inspace and time.

• Study on the exploitation of general learning functionalities: Apart from spectrumsensing results, large performance gains are expected for the SUs by using the re-sults from more general learning functionalities, e.g., the traffic-intensity modelof the PUs and transmission adaptation patterns used by the PTs. Recent re-search on certain functionalities has been very active, for example, spectrumanalysis, spectrum decision, and spectrum sharing. We envision large potentialsin the joint design of these functionalities for CR systems.

130 Chapter 7. Conclusions and Outlook

Appendix A

Mathematical Derivations

A.1 Solutions of [µi]k and [Σi]k,k′

The kth element of µi is

[µi]k = E (yk|Hi) =

∫xk,gk,yk

yk p (yk|xk, gk,Hi) p (xk, gk|Hi) dykdgkdxk

=

∫xk

p (xk|Hi)

∫gk

p (gk)

∫yk

yk p (yk|xk, gk,Hi) dykdgkdxk

=

∫xk

p (xk|Hi)

∫gk

gkxk p (gk) dgkdxk

= gk xk,i. (A.1)

where gk = E (gk) and xk,i = E (xk|Hi).The kth row and k

′th column of the covariance matrix is

[Σi]k,k′ =[E{

yyH|Hi

}− µiµ

Hi ]]

k,k′

=

∫yk,y

k′(yk − [µi]k)

(yk′ − [µi]k′

)∗ p(yk, yk′ |Hi

)dykdyk′

(a)=

∫yk,y

k′(yk − [µi]k)

(yk′ − [µi]k′

)∗ ∫xk,x

k′,gk,gk′

p(yk, yk′ |xk, xk′ , gk, gk′ ,Hi

)p(xk, xk′ , gk, gk′ |Hi

)dxkdxk′dgkdgk′dykdyk′

(b)=

∫xk,x

k′p(xk, xk′ |Hi

) ∫gk,g

k′p(gk, gk′

)∫

yk,yk′(yk − gkxk + gkxk − [µi]k)

(yk′ − gk′ xk′ + gk′ xk′ − [µi]k′

)∗p(yk, yk′ |xk, xk′ , gk, gk′ ,Hi

)dykdyk′dgkdgk′dxkdxk′

(c)=

∫xk,x

k′p(xk, xk′ |Hi

) ∫gk,g

k′p(gk, gk′

)∫

yk,yk′

(vkv∗

k′+(gkxk − [µi]k

) (gk′ xk′ − [µi]k′

)∗)p(yk, yk′ |xk, xk′ , gk, gk′ ,Hi

)dykdyk′dgkdgk′dxkdxk′

= σ2v δk,k′ + E {xkx∗k′}E {gkg∗k′} − [µi]k [µi]

∗k′

(A.2)

131

132 Appendix A. Mathematical Derivations

where step (a) uses the Bayesian rule to expand the conditional probability p(yk, yk′ |Hi)as a function of xk, xk′ , gk, and gk′ . Step (b) expresses p(xk, xk′ , gk, gk′ |Hi) as the prod-uct of p(xk, xk′ |Hi) and p(gk, gk′ ) by exploiting the mutual independency between thereceived signals and the reporting channels. The next step (c) simplifies the third inte-gral in the last step by considering that yk − gkxk = vk for all k and removes the itemsincluding the first order of vk and v

′k because theirs means are both equal to zero.

A.2 Analytical Form of Qavg(v) and Q′avg(v)

The key to obtain closed-form expression of Qavg(v) and Q′avg(v) is to derive

Q1(T) =∫ T

0f (t)dt, (A.3)

Q2(T) =∫ T

0t f (t)dt. (A.4)

Recalling that t = (σ2s + P1g12)/g22, the variables g22 and P1g12 follow an exponential

distribution with the mean l−122 and P1/l12, respectively, we follow a similar approach

as in [12, (48)] and derive the probability density distribution function of t as

f (t) =

l22l2

yσ2s + l22ly +

l222lyσ2

st

(l22 + lyt)2 e

−l22σ2s

t

t ≥ 0

0 t < 0

(A.5)

with ly = l21/P1. Taking (A.5) into (A.3), we have

Q1(T)(a)= a

∫ ∞

γ

z + a + 1(z + a)2 e−zdz

= a(∫ ∞

γ

1(z + a)

e−zdz +∫ ∞

γ

1(z + a)2 e−zdz

)(b)= a

(eaE1(γ + a) +

1γ + a

e−γ − eaE1(γ + a))

=a

γ + ae−γ.

where in step (a) we use the following replacements to simplify the integral:

a =l12

P1σ2

s , γ =l22

Tσ2

s , z =l22

tσ2

s .

In step (b), we use the definition of the exponential integral [4]

E1(x) =∫ ∞

x

e−t

tdt, R(x) > 0.

A.3. Proof of Proposition 4.2.2 133

Considering Q2(T), we take (A.5) into its expression and get

Q2(T)(a)= a

∫ ∞

γ

z + g + 1z(z + g)2 e−zdz

= a∫ ∞

γ

((1z− 1

z + g

)1g+

1g2

(− 1

z + g− g

(z + g)2 +1z

))e−zdz

=(g

a+

ga2

)E1(γ)−

ga2 eaE1(γ + a)− g

a(γ + a)e−γ

where in step (a), we use

g =l12l22

P1σ4

s .

Consequently, Qavg(v) and Q′avg(v) in (4.18) and (4.20) can be represented as

Qavg(v) = (PS − v)Q1(v− PS) + vQ1(v)− (Q2(v)−Q2(v− PS)) (A.6)

Q′avg(v) = Q1(v)−Q1(v− PS). (A.7)

A.3 Proof of Proposition 4.2.2

We first define the function r(T, α, β) with α + β/T > 0 which will be used later as

r(T, α, β) =

∫ T

0ln(

α +β

t

)f (t)dt. (A.8)

It is easy to see that r(T, α, β) = 0 for T < 0. In the following, we express theachievable rate as the function of r(T, α, β). Specifically, integrating the optimal powercontrol solution (4.10) into the objective function of (4.8), we obtain

R =

∫ v?−PS

0ln(

1 +PS

t

)f (t)dt +

∫ v?

v?−PS

ln(

v?

t

)f (t)dt. (A.9)

Using (A.9), after some simple mathematical computations, we can rewrite (A.9) as(4.22) for PS ≤ PI l21. Similarly, the performance is given in (4.23) for PS > PI l21.

Therefore, the key point to derive Proposition 4.2.2 is to calculate the closed-formexpression of r(T, α, β), T ≥ 0. Specifically, we take the expression of f (t) in (A.5) into(A.9) and obtain

r(T, α, β) =

∫ T

0ln(

α +β

t

)l22(l12/P1)

2σ2S + l22l12/P1 +

l222l12σ2

S/P1t

(l22 + l12t/P1)2 e

(−l22σ2

St

)dt

(a)=

∫ ∞

γb ln(α + x)

x + h + 1/g(x + h)2 e−gxdx

= b∫ ∞

γln(α + x)

(1

x + h+

1/g(x + h)2

)e−gxdx (A.10)


where, in (a), we use the following notation for brevity

γ =β

T, x =

β

t, b =

l12σ2S

P1, g =

l22σ2S

β, h =

l12β

l22P1(A.11)

Similar to [12, Eq. (54)], we consider the following integration∫ ∞

γln(α + x)

1x + h

e−gxdx

(a)=

1g

(ln(α + γ)

1γ + h

e−gγ −∫ ∞

γln(α + x)

1(x + h)2 e−gxdx +

∫ ∞

γ

1(x + h)(x + α)

e−gxdx

)

where we use integration by parts in (a). Hence, equation (A.10) is equal to

r(T, α, β) =bg

(ln(α + γ)

1γ + h

e−gγ +

∫ ∞

γ

1(x + h)(x + α)

e−gxdx

). (A.12)

Different from [12, Appendix A], we calculate r(T, α, β) in closed form by distinguish-ing between two cases. Particularly, For the cases h 6= α and h = α, we derive from(A.12) and have the expression of r(T, α, β) given in

r(T, α, β) =bg

(ln(α + γ)

1γ + h

e−gγ +

∫ ∞

γ

(1

x + h− 1

x + α

)1

α− he−gxdx

)=

bg

(ln(α + γ)

1γ + h

e−gγ +1

α− h

(eghE1(g(h + γ))− egαE1(g(α + γ))

)),

h 6= α (A.13)

r(T, α, β) =bg

(ln(α + γ)

1γ + h

e−gγ +

∫ ∞

γ

1(x + α)2 e−gxdx

)=

bg

(ln(α + γ)

1γ + α

e−gγ + e−gγ 1γ + α

− gegαE1(g(γ + α))

), h = α.

(A.14)

A.4 Monotonicity of G(x)

Calculating the derivative G(x) over x results in G′(x) = exE1(x)− 1/x. Now consider

the following inequality holds:

1x + 1

(a)< exE1(x)

(b)< ln

(1 +

1x

)(c)<

1x∀x > 0 (A.15)

where we apply [4, Eq. 5.1.20] in (a) and (b). In (c), we use ln(1 + x) < x, x >0. Finally, we conclude that G

′(x) is negative and G(x) is strictly monotonically

decreasing w.r.t. x.

A.5. Monotonicity of H(x) 135

A.5 Monotonicity of H(x)

Calculating the derivative of H(x) over x and rearranging the resulting terms, weobtain

dH(x)dx

= b1

(ea1+b1PSxE1(a1 + b1PSx)

+ b1PSx(

ea1+b1PSxE1(a1 + b1PSx)− 1a1 + b1PSx

))(a)= b1

((y + 1)

(eyE1(y)−

1y + 1

)− a1

(eyE1(y)−

1y

))(A.16)

where, in (a), we use y = a1 + b1PSx and rearrange the terms. Applying (A.15), thederivative in (A.16) is positive. Hence, we conclude that H(x) is a monotonicallyincreasing function of x.

Moreover, applying (A.15) to H(x) yields b1x/(a1 + b1PSx+ 1) < H(x) < b1x/(a1 +b1PSx). Using this inequality and the positivity of a1, b1, and PS, it is straightforwardto see that both the lower and upper bounds of H(x) converge to zero when x ap-proaches zero, thus we have limx→0 H(x) = 0.


If PS > PI l21, the finite and positive v?2 is chosen to satisfy Eg22{P?2 (g22)} = PI l21. We

first reformulate Eg22{P?2 (g22)} as a function of v?2 .

For the case v?2 > PS, integrating (4.60) into Eg22{P?2 (g22)} yields

Eg22{P?2 (g22)} =

∫ a1+1

b1(v?2−PS)a1+1b1v?2

p?(x) fg22(x)dx + PS

∫ ∞

a1+1

b1(v?2−PS)

fg22(x)dx

= PSe− (a1+1)l22

b1(v?2−PS) + v?2

(e− (a1+1)l22

b1v?2 − e− (a1+1)l22

b1(v?2−PS)

)

− (a1 + 1)l22

b1

(E1

((a1 + 1)l22

b1v?2

)− E1

((a1 + 1)l22

b1(v?2 − PS

)))(a)= v?2e

− λbv?2 − λbE1

(λbv?2

)−((v?2 − PS) e

− λbv?2−PS − λbE1

(λb

v?2 − PS

))(A.17)

where in (a), we introduce

λb =(a1 + 1)l22

b1. (A.18)


Similarly, for the case 0 < v?2 ≤ PS, we integrate (4.61) into Eg22{P?2 (g22)} and have

Eg22{P?2 (g22)} =

∫ ∞

a1+1b1v?2

p?(x) fg22(x)dx = v?2e− λb

v?2 − λbE1

(λbv?2

). (A.19)

Using the definition of F2(x) in (4.63), we combine the expressions in (A.17) and (A.19)in the general form Eg22{P?

2 (g22)} = F2(v?2).Therefore, in order to choose v?2 satisfying Eg22{P?

2 (g22)} = PI l21, we need tochoose v?2 as the root x? of F2(x) = PI l21. Note that x is used instead of v2 here.

Applying the Newton method to root-searching for F2(x) = PI l21 results in

x(n+1) = x(n) −F2

(x(n)

)− PI l21

F′2(x(n)). (A.20)

According to local convergence theory in [67, Theory 1.1], the standard assump-tions for the convergence of (A.20) are [67, Assumption 1.2.1]

1. F2(x) = PI l21 has a solution x?.

2. F′2(x) is Lipschitz continuous near x?.

3. F′2(x?) is nonsingular.

In the following, we prove that the sequence generated by (A.20) satisfies the standardassumptions.

Before proceeding to the proof, we first provide some useful expressions. Thefirst, second, and the third derivative of f2(x) in (4.64) w.r.t. x are given as

f′2(x) = e

−λbx (A.21)

f′′2 (x) = e

−λbx

λbx2 (A.22)

f′′′2 (x) = e

−λbx x−4λb(λb − 2x) (A.23)

and f′2(x) = f

′′2 (x) = f

′′′2 (x) = 0, x ≤ 0.

Considering the first condition, we first check

F2(0)− PI l21 = −PI l21 < 0 (A.24)

and we have

limx→∞

F2(x)− PI l21 = limx→∞

( f2(x)− f2(x− PS))− PI l21

(a)> lim

x→∞PS f

′2(x− PS)− PI l21

(b)= PS − PI l21

(c)> 0 (A.25)

where, in (a), we use the property that f2(x) is convex due to the positivity of f′′2 (x)

in (A.21). In (b), we use limx→∞ f′(x) = 1. The last step (c) uses the property that in


this case the interference power constraint is stricter than the peak power constraint,i.e., PS > PI l21. Furthermore, we check the first derivative of F2(x):

F′2(x) = f

′2(x)− f

′2(x− PS)

(a)> 0 (A.26)

where in (a), we use the property that f′2(x) is a monotonically increasing function due

to the positivity of f′′2 (x) in (A.21). From (A.26) it can be seen that F2(x) is a strictly

monotonically increasing function of x. Combining the conditions that F2(0)− PI l21and limx→∞ F2(x) − PI l21 have opposite signs, we conclude that the first conditionholds and the solution is unique.

Considering the second condition, we recall that an everywhere differentiablefunction is Lipschitz continuous if and only if its first derivative is bounded. There-fore, the second condition holds if we prove |F′′2 (x)| =

∣∣∣ f ′′2 (x)− f′′2 (x− PS)

∣∣∣ is bounded.

The proof is as follows. According to f′′′2 (x) in (A.21), the function f

′′2 (x) is monotoni-

cally increasing when 0 < x < λb/2 and monotonically decreasing afterwards. There-fore, f

′′2 (x) is a unimodal function and has the maximum at x = λb/2. Consequently,

|F′′2 (x)| < 2 f′′2 (x)

∣∣∣x=λb/2

=8

λbe2 < ∞. (A.27)

The third condition follows directly from F′2(x) 6= 0 according to (A.26).

So far, we have completed the proof that (A.20) satisfies the standard assumptions.The convergence thus depends on the selection of the initial point x(0). Herein, theapproximation of the root of F2(x) = PI l21 is provided as x(0). Specifically, using

the convexity of f (x), we have F2(x) = f2(x)− f2(x − PS) < PS f′2(x) = PSe−

λbx . Let

PSe− λb

x(0) = PI l21, we obtain x(0) in the form of (4.66) with λb = (a1 + 1)l22/b1.


Replacing P2(g22) in (4.32) with P?2 (g22) in (4.60) and (4.61), we have an expression of

the achievable performance of the suboptimal strategy DT-WF. We treat the cases thatP2(g22) is independent or dependent of g22 separately by defining the following twofunctions

r1(λ, p) =∫ ∞

λ

(ln(

1 +P2(x)x

σ2s

)+ el12s1 E1(l12s1)− el12s0 E1(l12s0)

)fg22(x)dx

∣∣∣∣∣P2(x)=p

(A.28)

r2(λ) =

∫ ∞

λ

(ln(

1 +P2(x)x

σ2s

)+ el12s1 E1(l12s1)− el12s0 E1(l12s0)

)fg22(x)dx

∣∣∣∣∣P2(x)= p?(x)

(A.29)


with nonnegative λ and p. Note that s1 is a function of P2(x). After some mathemati-cal manipulations, we obtain

RDT-WF =

{r1 (g1, PS)− r2 (g1) + r2 (g0) , v?2 > PS

r2 (g0) , 0 < v?2 ≤ PS.(A.30)

In order to get RDT-WF in closed form, we derive r1(λ, p) and r2(λ) in closed form.First, we decompose r1(λ, p) into three parts and evaluate them individually

r1(λ, p) = r11(λ, p) + r12(λ, p)− r13(λ) (A.31)

where r11(λ, p) is given as

r11(λ, p) =∫ ∞

λln(

1 +pxσ2

s

)l22e−l22xdx

(a)= ln

(1 +

pλ

σ2s

)e−l22λ + e

l22σ2s

p E1

(l22

(λ +

σ2sp

)). (A.32)

and in (a), we use integration by parts.The second term r12(λ, p) is represented as

r12(λ, p) =∫ ∞

λel12s1 E1(l12s1)l22e−l22xdx

(a)=

∫ ∞

λe

l12

(σ2

s +pxP1

)E1

(l12

(σ2

s + pxP1

))l22e−l22xdx

(b)=

∫ ∞

λea2+b2xE1 (a2 + b2x) l22e−l22xdx

(c)=

l22

b2e

l22a2b2

∫ ∞

a2+b2λe(

1− l22b2

)tE1(t)dt

(d)=

l22

b2e

l22a2b2 f3

(a2 + b2λ, 1− l22

b2

)(A.33)

where, in (a), we take P2(x) = p into s1 defined in (4.31), respectively. In (b), we use

a2 =l12σ2

sP1

(A.34)

b2 =l12pP1

. (A.35)

We use t = a2 + b2x in (c). In (d), f3(θ, ρ) is exploited which is defined and calculatedusing integration by parts as follows:

f3(θ, ρ) =

∫ ∞

θeρxE1(x)dx, θ ≥ 0, ρ < 1 (A.36)


=

1ρ

(−eθρE1(θ) + E1((1− ρ)θ)

), ρ < 1 and ρ 6= 0

e−θ − θE1(θ), ρ = 0.(A.37)

Using the property that s0 = σ2s /P1 is independent of g22 according to (4.31), the

third term is

r13(λ) =

∫ ∞

λel12s0 E1(l12s0)l22e−l22xdx = e

l12σ2s

P1−l22λE1

(l12σ2

sP1

)(A.38)

Second, we similarly decompose r2(λ) into three parts and evaluate them sepa-rately:

r2(λ) = r21(λ) + r22(λ)− r23(λ). (A.39)

Unlike P2(g22) in r1(λ, p), P2(g22) is a function of g22 in r2(λ).Particularly, the first term r21(λ) can be reformulated as

r21(λ)(a)=

∫ ∞

λln

1 +

(v?2 −

a1+1b1x

)x

σ2s

l22e−l22xdx

=

∫ ∞

λln(− P1

σ2s l12

+v?2σ2

sx)

l22e−l22xdx

(b)= ln

(− P1

σ2s l12

+v?2λ

σ2s

)e−l22λ + e

− P1 l22v?2 l12 E1

(l22

(λ− P1

v?2 l12

))(A.40)

where, in (a), we use the definition of p?2(g22) in (4.59). In (b), we use integrationby parts. According to the definition of the exponential integral, the real part of theargument of E1(·) should be positive. Therefore, we require that λ > P1/(v?2 l12) in(A.40). This is validated by noting that in this paper λ is chosen to be either g1 or g0as shown in (4.69) and it is easy to verify that g1 > g0 > P1/(v?2 l12).

The second term r22 is stated as

r22(λ) =

∫ ∞

λel12s1 E1(l12s1)l22e−l22xdx

(a)=

∫ ∞

λe

l12

(− 1

l12+

v?2P1

x)

E1

(l12

(− 1

l12+

v?2P1

x))

l22e−l22xdx

(b)=

∫ ∞

λea3+b3xE1 (a3 + b3x) l22e−l22xdx

(c)=

l22

b2e

l22a3b3 f3

(a3 + b3λ, 1− l22

b3

)(A.41)

where, in (a), we take the definition of p?(g22) in (4.59) into s1 in (4.31). In (b), we usea3 = −1 and b3 = v?2 l12/P1. Step (c) is similar to (c) and (d) in (A.33).


Lastly, due to the independence of s0 and g22, it straightforwardly follows that

r23(λ) = r13(λ) = el12σ2

sP1−l22λE1

(l12σ2

sP1

)(A.42)

To summarize, in (4.69), r1(λ, p) is given by (A.31) in which r11(λ, p), r12(λ, p),and r13(λ, p) are provided in (A.32), (A.33), and (A.38), respectively. Similarly, r2(λ, p)is expressed in (A.39) in which r21(λ, p), r22(λ, p), and r23(λ, p) are shown in (A.40),(A.41), and (A.42), respectively.

A.8 Strong Duality for the Optimization Problem (4.93)

According to [138, Theorem 1], [127, Theorem 4.1], if an optimization problem satisfiesboth the time-sharing property [138, Definition 1] and the Slater regularity condition[111], the primal and the dual problem have zero duality gap. Thus, in order to provestrong duality for (4.93), we investigate these two conditions separately.

1) Time-sharing property: We assume P?2 (t) and P?

2 (t) are optimal solutions to (4.93)with ε equal to ε and ε, respectively. Similar to [138], we choose the solution P2(t)which is equal to P?

2 (t) for α of the channel realizations and equal to P?2 (t) for 1− α of

the channel realizations. It is straightforward to show that P2(t) is a feasible solutionto (4.93) with

εp = αεp + (1− α)εp

and that the following inequality

R(P2(t)

)≥ αR

(P?

2 (t))+ (1− α)R

(P?

2 (t))

(A.43)

is satisfied with equality. The term R(·) is defined in (4.9).2) Slater regularity condition: For (4.93), the Slater regularity condition requires the

existence of a solution in the relative interior of the domain that is strictly feasible[111]. It is easy to show that this condition holds for any ε ∈ [0, 1) since the solutionP?

2 (t) = δ, δ ∈(

0, 1a (

1ε − 1)

), ∀t, always satisfies all constraints with strict inequality.


The Lagrangian of (4.93) is written as

L = Et

{ln(

1 +P2(t)

t

)}+ λ

(Et

{1

1 + aP2(t)

}− ε

)(A.44)

where the non-negative Lagrangian multiplier λ corresponds to the constraint (4.91).Herein, no Lagrange multiplier is assigned to the peak power constraints since theyare considered in the generalized KKT conditions as shown later.

A.10. Feasible Region Ft for x1,2(t, v) 141

According to the generalized KKT conditions for functional optimization [85, 87],if the optimal solution P?

2 (t) is a regular point, it satisfies the following conditions

dldP2(t)

∣∣∣∣∣P2(t)=P?

2 (t)

= 0, 0 < P?

2 (t) < PS

≤ 0, P?2 (t) = 0

≥ 0, P?2 (t) = PS

(A.45)

λ? ≥ 0 (A.46)

λ?

(Et

{1

1 + aP?2 (t)

}− ε

)= 0 (A.47)

Et

{1

1 + aP?2 (t)

}≥ ε (A.48)


l = ln(

1 +P2(t)

t

)+ λ?

(1

1 + aP2(t)− ε

)(A.49)

and the derivative of (A.45) with respect to (w.r.t.) P2(t) is

dldP2(t)

=1

t + P2(t)− λ? a

(1 + aP2(t))2 . (A.50)

By solving the KKT conditions (A.45), (A.46), and using v = 1/λ, we obtain thesolutions as given in (4.95), (4.96), and (4.97).

The KKT conditions (A.47) and (A.48) are used to determine the value of v?.Particularly, if the outage constraint is inactive at the optimal solution, v? should beinfinity since λ? = 0 according to (A.47). We prove by contradiction that under thiscircumstance, the optimal solution is P?

2 (t) = PS for all t. If P?2 (t) 6= PS for some

t, the instantaneous transmit power could be increased to improve the secondarytransmission rate while still satisfying the outage constraint. It can be seen from(A.48) that this case happens if ε ∈ [0, 1/(1 + aPS)). Furthermore, if ε = 1/(1 + aPS),it is easy to see that the optimal solution is also P?

2 (t) = PS, thus the calculation of v?

is not required. Therefore, we only focus on calculating v? if 1/(1 + aPS) < ε < 1. Inthis case, v? should be chosen to let the constraint Qc(P?

2 (t)) = ε hold.

A.10 Feasible Region Ft for x1,2(t, v)

The feasible region Ft is the set of t to let the solutions x1,2(t, v) be a real number with0 < x1,2(t, v) < PS. In order to guarantee that x1,2(t, v) is real, the non-negativity ofthe part under the square root in (4.98) and (4.99) is required. This yields

t ≥ T3 (v) , T3 (v) =4v− 1

4av. (A.51)


In the following, we characterize the feasible regions of t for x1(t, v) and x2(t, v)separately. Firstly, considering x1(t, v), we distinguish between three cases:

1. 1− 2v ≤ 0: Calculating the region of t to let x1(t, v) satisfy 0 < x1(t, v) < PS, wehave

T2 (v) < t < T1 (v) (A.52)

where

T1(v) =(1 + aPS)

2

av− PS, T2(v) =

va

, (A.53)

and T1(v) > T2(v). Additionally, we have T2 (v) ≥ T3 (v) since

T2 (v)− T3 (v) =1 + 4v2 − 4v

4av=

(1− 2v)2

4av≥ 0 (A.54)

which shows that (A.52) implies the condition (A.51).

2. 0 < 1− 2v < 2avPS: Given that the condition in (A.51) holds, the positivity ofx1(t, v) is straightforwardly satisfied. We calculate the region of t to let x1(t, v) ∈(0, PS) hold and obtain t < T1 (v). Therefore, the feasible region of t is

T3 (v) ≤ t < T1 (v) (A.55)

where we use the relation T1 (v) ≥ T3 (v) due to

T1 (v)− T3 (v) =(2v (aPS + 1)− 1)2

4av≥ 0. (A.56)

3. 1− 2v ≥ 2avPS: It is easy to verify that x1(t, v) is not the feasible solution forany non-negative t since it will always be equal to or exceed PS.

Secondly, we consider the solution x2(t, v). The term 1− 2v should be positive inorder to satisfy the positivity of x2(t, v). We classify the discussion into two cases:

1) 0 < 1− 2v < 2avPS: If the condition in (A.51) holds, x2(t, v) is real and x2(t, v) <PS. To restrain x2(t, v) to be positive, we have t < T2(v). Therefore, the feasible regionof t is

T3 (v) ≤ t < T2 (v) . (A.57)

2) 1− 2v ≥ 2avPS: We obtain the region of t such that 0 < x2(t, v) < PS is satisfiedas

T1 (v) < t < T2 (v) (A.58)

where we use T1 (v) < T2 (v) for 1− 2v > avPS. Using the inequality T1(v) ≥ T3(v),cf. (A.56), we show that (A.58) implies the satisfaction of the condition (A.51).

In summary, the region of t to make x1(t, v) feasible is (A.52) for 1− 2v ≤ 0, and(A.55) for 0 < 1− 2v < 2avPS. The region of t to make x2(t, v) feasible is (A.57) for0 < 1− 2v < 2avPS and (A.58) for 1− 2v ≥ 2avPS.

A.11. Proof of Lemma 4.3.1 143

A.11 Proof of Lemma 4.3.1

Consider that for the case 1/(1+ aPS) < ε < 1, the outage probability constraint holdswith equality at the optimum and v? is a positive finite value. Since the consideredfunctions are in twice differentiable class, according to [58, Chapter 7.2, Proposition1], a sufficient condition for x?(t) to yield a local maximum of (4.93) is that x?(t) is aregular point and satisfies

Lxx (x?(t), λ?; ξ(t)) < 0. (A.59)

where Lxx(·; ·) is the second-order functional derivative of the Lagrange function L in(A.44) w.r.t. x in the direction ξ(t). Similarly, a sufficient condition for x?(t) yieldinga local minimum of (4.93) is Lxx (x?(t), λ?; ξ(t)) > 0.

Without loss of generality, we focus on checking the sufficient conditions for a lo-cal maximum unless specified otherwise. Specifically, defining Q

′c (x(t)) as the func-

tional derivative of Qc(x(t)) w.r.t. x(t) at x?(t) [34], ξ(t) is any vector of the kernel ofthe mapping Q

′c(x?(t)). Incorporating (A.44) into the left side of (A.59), we have

Lxx (x?(t), λ?; ξ(t)) =d2

dϕ2 L (x?(t) + ϕξ(t), λ?)∣∣∣

ϕ=0

=

∫ ∞

0

(− 1

(t + x?(t))2 +2λ?a2

(1 + ax?(t))3

)f (t)ξ2(t)dt (A.60)

with f (t) representing the PDF of t. Integrating (A.60) into (A.59), it yields

G(x?(t)) = − 1

(t + x?(t))2 +2λ?a2

(1 + ax?(t))3 < 0. (A.61)

However, verifying the optimality of x1(t, v?) and x2(t, v?) by checking (A.61) isdifficult since incorporating them into (A.61) yields involved expressions. Alterna-tively, we rewrite (A.61) into equivalent form that is simple to verify. Specifically,using λ = 1/v, we construct

x1,2(t, λ) =λ− 2±

√λ2 − 4λ + 4aλt

2a(A.62)

and it is easy to verify that

x1(t, λ) = x1(t, v), x2(t, λ) = x2(t, v).

Consequently, x1(t, λ?) and x2(t, λ?) are roots of dl/dP2(t) = 0, where dl/dP2(t)was given in (A.50). Thus, the equation

H(x)∣∣∣x=x1,2(t,λ?)

=(1 + ax)2

t + x

∣∣∣x=x1,2(t,λ?)

=λ?a (A.63)


follows directly. With the help of (A.63), G(x?(t)) in (A.61) at x1,2(t, λ?) is

G(x1,2(t, λ?))(a)= − 1

(t + x1,2(t, λ?))2 +1

t + x1,2(t, λ?)

2a(1 + ax1,2(t, λ?))

=1

(1 + ax1,2(t, λ?))2

(− (1 + ax1,2(t, λ?))2

(t + x1,2(t, λ?))2 +2a(1 + ax1,2(t, λ?))

t + x1,2(t, λ?)

)(b)=

1(1 + ax1,2(t, λ?))2 H′(x1,2(t, λ?)) (A.64)

where, in (a), we incorporate (A.63) and, in (b), we insert H′(x) at x1,2(t, λ?). Com-

paring (A.64) with (A.61), we conclude that in order to satisfy the sufficient condition(A.61) for a local maximum point, we can equivalently check whether H′(x)|x=x1,2(t,λ?) <0. This condition combined with H(x1,2(t, λ?)) = λ?a implies that if x1,2(t, λ?) is astrictly monotonically decreasing function of λ?, then x1,2(t, λ?) is a local maximumpoint. Analogously, if x1,2(t, λ?) is a strictly monotonically increasing function of λ?,then x1,2(t, λ?) is a local minimum point.

This equivalent sufficient condition is verified as follows. Taking the derivative ofx1,2(t, λ?) in (4.98) and (4.99) w.r.t. to λ? yields

dx1,2(t, λ?)

dλ?=

12a

(1± λ? − 2 + 2at√

(λ?)2 − 4λ? + 4atλ?

)(A.65)

We use the property that the term under the square root should be non-negative, i.e.,(λ?)2 − 4λ? + 4atλ? ≥ 0, we thus have t ≥

(4λ? − (λ?)2)/(4aλ?). Consequently, we

have

λ? − 2 + 2at ≥ λ? − 2 +4λ? − (λ?)2

2λ?=

λ?

2

(a)> 0 (A.66)

where, in (a), we use that herein λ? is positive. Taking it into (A.65), for x?1(t)

dx1(t, λ?)

dλ?=

12a

(1 +

λ? − 2 + 2at√(λ?)2 − 4λ? + 4atλ?

)> 0 (A.67)

For x?2(t), we obtain

dx2(t, λ?)

dλ?

(a)<

12a

(1− λ? − 2 + 2at

λ? − 2 + 2at

)= 0 (A.68)

where, in (a), we use the positivity of λ? − 2 + 2at in (A.66) and the inequality

(λ?)2 − 4λ? + 4atλ? < (λ? − 2 + 2at)2 (A.69)

for at 6= 1. Note that we do not consider the case when at = 1 since incorporating itinto (A.62) yielding x2(t, λ?) = −1/a, which is negative and infeasible.

The inequalities (A.67) and (A.68) imply that x1(t, λ?) and x2(t, λ?)) are strictlymonotonically increasing and decreasing functions of λ?, respectively. Therefore,

A.12. Proof of Lemma 4.3.2 145

x1(t, v) is a local minimum solution and x2(t, v) is a local maximum solution in thecorresponding region of t.

A.12 Proof of Lemma 4.3.2

In this Appendix, we use x?(t) to denote the optimal solution of (4.93). The lemmais proved by reduction to absurdity. Specifically, assuming there exists a t1 and a t2with t1 < t2 such that x?(t) in the vicinity of t1 and t2 is continuous and values ofx?(t) in the vicinity of t1 are strictly smaller than those in the vicinity of t2. In theremaining region of t, x?(t) is assumed to be decreasing w.r.t. t. We choose small δ1and δ2 satisfying t1 + δ1 < t2 − δ2 and

A =

∫ t1+δ1

t1−δ1

f (t)dt =∫ t2+δ2

t2−δ2

f (t)dt.

According to the first mean value theorem [52] and using the property that the pdff (t) is bounded, there exist t1 and t2 such that∫ t1+δ1

t1−δ1

11 + ax?(t)

f (t)dt =1

1 + ax?(t1)A∫ t2+δ2

t2−δ2

11 + ax?(t)

f (t)dt =1

1 + ax?(t2)A.

Note that x?(t1) < x?(t2). We can construct a new solution

x?(t) =

x?(t2), if t ∈ [t1 − δ1, t1 + δ1]

x?(t1), if t ∈ [t2 − δ2, t2 + δ2]

x?(t), otherwise

. (A.70)

The solution x?(t) yields the same outage probability for the PU as x?(t). Therefore,it also satisfies the outage probability constraint. Calculating the difference betweenthe secondary achievable rate with x?(t) and that with x?(t), we obtain

limδ1, δ2→0

R(x?(t))− R(x?(t))

= limδ1→0

∫ t1+δ1

t1−δ1

(ln(

1 +x?(t2)

t

)− ln

(1 +

x?(t)t

))f (t)dt

+ limδ2→0

∫ t2+δ2

t2−δ2

(ln(

1 +x?(t1)

t

)− ln

(1 +

x?(t)t

))f (t)dt

(a)= lim

δ1, δ2→0A(

ln(

t1 + x? (t2)

t1 + x? (t1)

)+ ln

(t2 + x? (t1)

t2 + x? (t2)

))(b)=A

(ln(

t1 + x? (t2)

t1 + x? (t1)

)+ ln

(t2 + x? (t1)

t2 + x? (t2)

))(c)> 0 (A.71)


where, in (a), the first mean value theorem is applied. In (b), we use the fact thatlimδ1→0 t1 = t1 and limδ2→0 t2 = t2. The last step (c) is due to the a priori assumptionthat x?(t1) < x?(t2)

The inequality (A.71) states that there exists another solution x?(t) that is a de-creasing function of t that satisfies the outage constraint and results in a higher achiev-able rate than the solution x?(t). Therefore, the optimal solution of (4.93) is a decreas-ing function of t.


For the case 1− 2v ≥ 2avPS, we incorporate (4.106) into Qc(v) with x2(t, v) in (4.111)and have

Qc(v) =1

1 + aPS

∫ T1(v)

0f (t)dt +

1− 2va

∫ T2(v)

T1(v)

1(1− v)/a− t

f (t)dt +∫ ∞

T2(v)f (t)dt

Defining

Q1(T) =∫ T

0f (t)dt (A.72)

Q2(λ, T) =∫ T

0

1λ− t

f (t)dt (A.73)

with λ > T and nonnegative λ, we arrive at (4.113). A similar derivation is performedto generate (4.114), (4.115) and (4.116). Therefore, the key to derive a closed-formexpression of Qc(v) is to obtain the analytical form of Q1(T(v)) and Q2(T(v)). Notethat we use T instead of T(v) for notational brevity in the remainder of this proof.

It is easy to verify Q1(T) = 0 and Q2(T) = 0 for T < 0. In the following, we onlyconsider the case T ≥ 0. Taking the PDF of t (A.5) into (A.72) and (A.73), we have

Q1(T)(a)= b

∫ ∞

ρ

z + b + 1(z + b)2 e−zdz

= b

(∫ ∞

ρ

1(z + b)

e−zdz +∫ ∞

ρ

1(z + b)2 e−zdz

)(b)= b

(ebE1(b + ρ) +

1b + ρ

e−ρ − ebE1(b + ρ)

)=

bb + ρ

e−ρ.

where, in (a), we use the substitutions to simplify the integral

b =l12σ2

SP1

, ρ =l22σ2

ST

, z =l22σ2

St

(A.74)


In (b), we use the exponential integral E1(x) =∫ ∞

x e−t/tdt, R(x) > 0 [4].Similarly, we derive Q2(T) by taking (A.5) into its expression (A.73). After some

mathematical computations, we have

Q2(λ, T) =∫ T

0

1λ− t

f (t)dt

=bλ

∫ ∞

d

zz− c

z + b + 1(z + b)2 e−zdz

=bλ

∫ ∞

d

((c

b + c+

c(b + c)2

)1

z− c+

(b

b + c− c

(b + c)2

)1

z + b

+b

(b + c)(z + b)2

)e−zdz

=bλ

(c(b + c + 1)(b + c)2 e−cE1 (d− c) +

b(b + c)(b + d)

e−d

− c(b + c)2 ebE1 (b + d)

)

where c = l22σ2S

λ .


Applying the definition r(T, α, β) in (A.8) with α +β

T> 0 to the achievable rate ex-

pressions, e.g., for the case 1− 2v? ≥ 2av?PS, we incorporate (4.106) into (4.9) andhave

R(P?

2 (t))=

∫ T1(v?)

0ln(

1 +PS

t

)f (t)dt +

∫ T2(v?)

T1(v?)ln(−2v?

1− 2v?+

v?

(1− 2v?)at

)f (t)dt

(A.75)

after some simple mathematical computations, we can rewrite (A.75) as (4.117). Simi-larly, by incorporating (4.107), (4.109), and (4.110) into (4.9), we represent the achiev-able rate as a function of r(T, α, β) in (4.118), (4.119), and (4.120), respectively.

A.15 Solution of ∆Gk

We calculate the expression of ∆Gk in (5.65), for ∀ k = 1, . . . , K

∆Gk : = ∇∗k∑K

l=1EHl {MSEl} (A.76)


=∂∑K

l=1 Tr{

INl − GHl LHH

l B−1l HlLHGl

}∂G∗k

= −

∑Kl=1 Tr

{∂(GH

l )LHHl B−1

l HlLHGl

}∂G∗k︸︷︷︸

L1

+

∑Kl=1 Tr

{GH

l LHHl B−1

l ∂ (Bl) B−1l HlLHGl

}∂G∗k︸︷︷︸

L2

where L1 and L2 are the abbreviations of the two derivative terms. In the followingwe derive the detailed expressions of these terms. Specifically, for the first term

∂L1

∂[G∗k]

m,n

= Tr{

emeHn LHH

k BkHkLHGk

}= Tr

{eH

n LHHk BkHkLHGkem

}

⇒ ∂L1

∂G∗k= LHH

k BkHkLHGk (A.77)

For the second term, the derivative is given as

∂L2

∂[G∗k ]m,n

=K∑

l=1

Tr

{GH

l LHHl B−1

l

∂

( K∑i=1

HlLHGiGHi LHH

l + Tr{

LHGiGHi L(

Re,TxHi

)T}

Re,RxHi

σ2l I)

∂[G∗k ]mn

B−1l HlLHGl

}

=K∑

l=1

Tr

{GH

l LHHl B−1

l

(HkLHGkemeH

n LHHk + Tr

{LHGkemeH

n L(

Re,TxHk

)T}

Re,RxHk

)

B−1l HlLHGl

}

=K∑

l=1

(Tr{

eHn LHH

k B−1l HkLHGlGH

l LHHl B−1

l HkLHGkem

}+ Tr

{eH

n L(

Re,TxHi

)TLHGkem

}Tr{

GHl LHH

l B−1l Re,Rx

HkB−1

l HlLHGl

}). (A.78)

A.16. Solution of the Projection Operation 149

Thus, we have

∂L2

∂G∗k=

K∑l=1

(LHH

k B−1l HlLHGlGH

l LHHl B−1

l HkLHGk

+ Tr{

GHl LHH

l B−1l Re,Rx

HkB−1

l HlLHGl

}L(

Re,TxHk

)TLHGk

). (A.79)

Combining (A.77) and (A.79), we arrive at (5.65).

A.16 Solution of the Projection Operation

Consider the problem (5.66), both the objective function and the power constraint areconvex functions of G. Thus, solving the projection operation is equivalent to solvinga convex optimization problem. The Lagrange function of (5.66) is

L(G, µ) =K∑

k=1

∥∥∥Gk − Gk

∥∥∥2

F+ µ

( K∑k=1

Tr{

GkGHk

}− P

).

where µ is the Lagrange multiplier associated with the equivalent power constraint.According to KKT conditions, the following equality holds to derive the optimum

G?k − Gk + µG?

k = 0, ∀ k = 1, . . . , KK∑

k=1

Tr{

G?k(G?

k)H}− P = 0

Solving from the last two equations, we obtain

G?k =

√√√√√ PK∑

i=1

∥∥∥Gi

∥∥∥2

F

Gk.

A.17 Solution of gupp

Defining the function

F(g22) =p(H1) f (γ|H1)σ2

s +P1/l12g22

+ PS

+p(H0) f (γ|H0)

σ2s

g22+ PS

(A.80)

we note that it has the finite limit when g22 approaches infinity

limg22→∞

F(g22) =p(H0) f (γ|H0) + p(H1) f (γ|H1)

PS. (A.81)


Combined with (6.19), this indicates that if limg22→∞ F(g22) ≤ λ?2 f (γ|H1), then

gupp = ∞ (A.82)

which means that the feasible condition (6.19) for P?2 (γ, g22) = PS is always violated,

i.e., P?2 (γ, g22) can not reach the value PS. Herein, we only consider the finite g22.

If limg22→∞ F(g22) > λ?2 f (γ|H1), we try to solve gupp as the root of

F(x) = λ?2 f (γ|H1). (A.83)

After some mathematical calculation, we reformulate (A.83) as a quadratic equation

a1x2 + b1x + c1 = 0 (A.84)

with

a1 = PS (p(H0) f (γ|H0) + p(H1) f (γ|H1)− λ?2 f (γ|H1)PS) (A.85)

b1 = p(H1) f (γ|H1)σ2s + p(H0) f (γ|H0)

(σ2

s +P1

l12

)− λ?

2 f (γ|H1)PS

(2σ2

s +P1

l12

)(A.86)

c1 =− λ?2 f (γ|H1)

(σ4

s +P1

l12σ2

s

). (A.87)

Note that the preliminary condition limg22→∞ F(g22) > λ?2 f (γ|H1) implies a1 > 0. In

addition, c1 is negative and finite due to the positive and finite value of λ?2 .

The roots solving the quadratic equation (A.84) are

x1,2 =−b1 ±

√b2

1 − 4a1c1

2a1(A.88)

with x1x2 = c1/a1 < 0. Using (A.85)-(A.87), we can verify that√

b21 − 4a1c1 > |b1|.

Hence, we obtain gupp as the nonnegative solution among x1,2

gupp =−b1 +

√b2

1 − 4a1c1

2a1. (A.89)

To summarize, gupp is given as

gupp =

−b1 +

√b2

1 − 4a1c1

2a1, limg22→∞ F(g22) > λ?

2 f (γ|H1)

∞, otherwise.

(A.90)

Appendix B

Notation

a, b, . . . , A, B, . . . scalarsa, b, . . . vectorsdiag{a} square matrix: the elements of a are on the diagonal

and the other entries are all zeroA, B, . . . matricesA, B, C, . . . sets|A| cardinality of the set AN set of all natural numbersC set of all complex numbers[a]k kth element of the vector a[A]k,l element of the matrix A in the kth row and lth columnAT transpose of the matrix AAH conjugate transpose of the matrix AA∗ (element-wise) complex conjugate of the matrix Avec {A} vectorization, i.e., column-wise stacking, of the matrix Atr {A} trace of the square matrix Adet{A} determinant of the square matrix A‖A‖F Frobenius norm of the matrix Arank {A} rank of the matrix Aλmax(A) maximal eigenvalue of the Hermitian matrix AA

12 unique Hermitian positive semidefinite square root of the

Hermitian positive semidefinite matrix AA� B Hadamard (element-wise) product of the matrices A and BA⊗ B Kronecker product of the matrices A and BPr{A} probability of the event AIN N × N identity matrix0M,N M× N all-zero matrix0N N × N all-zero matrix1M,N M× N all-one matrixek kth standard basis for K-dimensional spacex(n) the value of x in the nth iteration∑

k x[k] sum (from −∞ to ∞ unless otherwise specified){xk}K

k=1 the set contains xk, k = 1, . . . , Ka? optimal value of aln(a) natural logarithm of alog2(a) logarithm of a to the base 2

151

152 Appendix B. Notation

log10(a) logarithm of a to the base 10(a)+ max(a, 0)Ex{y(x)} expectation of y(x) over xf′(x) first derivative of the function f (x) over x

f′′(x) second derivative of the function f (x) over x

f′′′(x) third derivative of the function f (x) over x

R(x) real part of xCN (µ, R) multivariate complex normal distribution of

a k-dimensional random vector with location parameter µ

and covariance matrix R

Acronyms

ALRT approximated likelihood ratio testAP access pointAWGN additive white Gaussian noiseBER bit error rateBPSK binary phase-shift keyingBS base stationCDF cumulative density functionCR cognitive radioCSI channel state informationDL-DA downlink-based dual-loop algorithmDoF degree of freedomDSA dynamic spectrum accessDT-WF double threshold waterfillingDTCP-WF double threshold constant-power waterfillingED energy detectionETSI European Telecommunications Standards InstituteFC fusion centerFCC Federal Communication CommitteeFDD frequency division duplexGLRT generalized likelihood ratio testi.i.d. independent and identically distributedIEEE Institute of Electrical and Electronics EngineersIT interference temperatureITU International Telecommunication UnionKKT Karush-Kuhn-TuckerLLF log-likelihood functionLLR log likelihood ratioLLRT log likelihood ratio testLMMSE linear minimum mean squared errorLRT likelihood ratio test

153

154 Acronyms

LTE long term evolutionMIMO multiple-input multiple-outputMISO multiple-input single-outputMLE maximum likelihood estimateMME maximum-minimum eigenvalueMMSE minimum mean squared errorMSE mean squared errorMT mobile terminalMUI multiuser interferenceNP Neyman-PearsonOFDM orthogonal frequency-division multiplexingPDF probability density functionPMF probability mass functionPR primary receiverPSD positive semidefinitenessPT primary transmitterPU primary userQoS quality of serviceROC receiver operating characteristicSDP semi-definite programmingSDR semi-definite relaxationSINR signal-to-interference-and-noise ratioSISO single-input single-outputSNR signal-to-noise ratioSOCP second-order cone programmingSR secondary receiverST secondary transmitterSU secondary userTDD time division duplexTDMA time division multiple accessTV televisionUHF ultra high frequencyUL–DL uplink–downlinkUL-DA uplink-based dual-loop algorithmUMP uniformly most powerfulUWB ultra-widebandVHF very high frequencyWRAN Wireless Regional Area Network

Bibliography

[1] “Report of the spectrum efficiency working group,” FCC Spectrum Policy TaskForce, Tech. Rep., Nov. 2002. [Online]. Available: http://www.fcc.gov/sptf/files/SEWGFinalReport_1.pdf

[2] “Unlicensed operation in the TV broadcast bands,” Second Memorandum Opinionand Order, Sep. 2010, FCC 10-174. [Online]. Available: https://federalregister.gov/a/2010-30184

[3] 802.22 Working Group, “IEEE draft standard for information technology telecommu-nications and information exchange between systems wireless regional area networks(WRAN)-specific requirements part 22: Cognitive wireless RAN medium access control(MAC) and physical layer (PHY) specifications: Policies and procedures for operationin the TV bands,” IEEE P802.22/D1.0, pp. 1–598, Dec 2010.

[4] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas,Graphs, and Mathematical Tables. Mineola, NY, USA: Dover Publications, 1965.

[5] I. F. Akyildiz, W. Y. Lee, M. C. Vuran, and S. Mohanty, “Next generation/dynamicspectrum access/cognitive radio wireless networks: A survey,” J. COMPUTER NETW.(ELSEVIER), vol. 50, pp. 2127–2159, 2006.

[6] I. F. Akyildiz, W. Y. Lee, M. C. Vuran, and S. Mohanty, “A survey on spectrum man-agement in cognitive radio networks,” IEEE Commun. Mag., vol. 46, no. 4, pp. 40–48,Apr.

[7] T. W. Anderson, “Asymptotic theory for principle component analysis,” The Annals ofMathematical Statistics, vol. 34, no. 1, pp. 122–148, Mar. 1963.

[8] V. Asghari and S. Aissa, “Adaptive rate and power transmission in spectrum-sharingsystems,” IEEE Trans. Wireless Commun., vol. 9, no. 10, pp. 3272–3280, Oct. 2010.

[9] E. Axell and E. G. Larsson, “Optimal and sub-optimal spectrum sensing of OFDM sig-nals in known and unknown noise variance,” IEEE J. Sel. Areas Commun., vol. 29, no. 2,pp. 290–304, Feb. 2011.

[10] E. Axell and E. G. Larsson, “Eigenvalue-based spectrum sensing of orthogonal space-time block coded signals,” IEEE Trans. Signal Process., vol. 60, no. 12, pp. 6724–6728, Dec.2012.

[11] E. Axell, G. Leus, E. G. Larsson, and H. V. Poor, “Spectrum sensing for cognitive radio:State-of-the-art and recent advances,” IEEE Signal Process. Mag., vol. 29, no. 3, pp. 101–116, May 2012.

155

http://www.fcc.gov/sptf/files/SEWGFinalReport_1.pdf

http://www.fcc.gov/sptf/files/SEWGFinalReport_1.pdf

https://federalregister.gov/a/2010-30184

https://federalregister.gov/a/2010-30184

156 BIBLIOGRAPHY

[12] A. Bagayoko, I. Fijalkow, and P. Tortelier, “Power control of spectrum-sharing in fadingenvironment with partial channel state information,” IEEE Trans. Signal Process., vol. 59,no. 5, pp. 2244–2256, May 2011.

[13] A. Ben-Tal and A. Nemirovski, Lectures on modern convex optimization: analysis, algorithms,and engineering applications. Philadelphia, PA, USA: MPS-SIAM Series on Optimization,2001.

[14] D. P. Bertsekas, Nonlinear Programming. Belmont, MA, USA: Athena Scientific, 2008.

[15] D. P. Bertsekas and J. N. Tsitsiklis, Parallel and distributed computations. EnglewoodCliffs, NJ, USA: Prentice Hall, 1989.

[16] P. Bianchi, J. Najim, G. Alfano, and M. Debbah, “Asymptotics of eigenbased collabora-tive sensing,” in Proc. IEEE Inf. Theory Workshop (ITW), Sicily, Italy, Oct. 2009.

[17] R. S. Blum, “MIMO capacity with interference,” IEEE J. Sel. Areas Commun., vol. 21,no. 5, pp. 793–801, June 2003.

[18] T. E. Bogale, B. K. Chalise, and L. Vandendorpe, “Robust transceiver optimization fordownlink multiuser MIMO systems,” IEEE Trans. Signal Process., vol. 59, no. 1, pp. 446–453, Jan. 2011.

[19] R. L. Burden and J. D. Faires, Numerical Analysis, 8th ed. Boston, MA, USA: BrooksCole, 2005.

[20] H. Celebi and H. Arslan, “Utilization of location information in cognitive wireless net-works,” IEEE Wireless Commun., vol. 14, no. 4, pp. 6–13, Aug. 2007.

[21] V. Chakravarthy, X. Li, Z. Wu, M. Temple, F. Garber, R. Kannan, and A. Vasilakos,“Novel overlay/underlay cognitive radio waveforms using SD-SMSE framework to en-hance spectrum efficiency- part I: Theoretical framework and analysis in AWGN chan-nel,” IEEE Trans. Commun., vol. 57, no. 12, pp. 3794–3804, 2009.

[22] V. Chakravarthy, X. Li, R. Zhou, Z. Wu, and M. Temple, “Novel overlay/underlay cogni-tive radio waveforms using SD-SMSE framework to enhance spectrum efficiency-part II:Analysis in fading channels,” IEEE Trans. Commun., vol. 58, no. 6, pp. 1868–1876, 2010.

[23] H. S. Chen, W. Gao, and D. G. Daut, “Signature based spectrum sensing algorithms forIEEE 802.22 WRAN,” in Proc. IEEE Int. Conf. Commun. (ICC), Glasgow, Scotland, June2007, pp. 6487–6492.

[24] CVX Research Inc., “CVX: Matlab software for disciplined convex programming,” http://cvxr.com/cvx, Sep. 2012.

[25] A. V. Dandawate and G. B. Giannakis, “Statistical tests for presence of cyclostationarity,”IEEE Trans. Signal Process., vol. 42, no. 9, pp. 2355–2369, Sep. 1994.

[26] F. F. Digham, M. S. Alouini, and M. K. Simon, “On the energy detection of unknownsignals over fading channels,” IEEE Trans. Commun., vol. 55, no. 1, pp. 21–24, Jan. 2007.

[27] M. Ding, “Multiple-input multiple-output wireless system designs with imperfect chan-nel knowledge,” Dissertation, Queen’s University, Kingston, Ontario, Canada, July 2008.

http://cvxr.com/cvx

http://cvxr.com/cvx

BIBLIOGRAPHY 157

[28] P. A. Dmochowski, H. A. Suraweera, P. J. Smith, and M. Shafi, “Impact of channelknowledge on cognitive radio system capacity,” in Proc. IEEE. Veh. Technol. Conf. (VTC-Fall), Ottawa, Canada, Sep. 2010.

[29] Y. C. Eldar, A. Ben-Tal, and A. Nemirovski, “Robust mean-squared error estimationin the presence of model uncertainties,” IEEE Trans. Signal Process., vol. 53, no. 1, pp.168–181, Jan. 2005.

[30] Y. C. Eldar and A. Nehorai, “Mean-squared error beamforming for signal estimation: Acompetitive approach,” in Robust Adaptive Beamforming. Hoboken, NJ, USA: John Wileyand Sons, 2006, pp. 259–298.

[31] S. Filin, H. Harada, H. Murakami, and K. Ishizu, “International standardization of cog-nitive radio systems,” IEEE Commun. Mag., vol. 49, no. 3, pp. 82–89, Mar. 2011.

[32] W. A. Gardner and C. M. Spooner, “Signal interception: Performance advantages ofcyclic-feature detectors,” IEEE Trans. Commun., vol. 40, no. 1, pp. 149–159, Jan. 1992.

[33] M. Gastpar, “On capacity under receive and spatial spectrum-sharing constraints,” IEEETrans. Inf. Theory, vol. 53, no. 2, pp. 471–487, Feb. 2007.

[34] I. M. Gelfand and S. V. Fomin, Calculus of Variations. Englewood Cliffs, NJ, USA:Prentice Hall, 1963, English version: Dover Publications, Mineola, NY, USA, 2000.

[35] E. A. Gharavol, Y. C. Liang, and K. Mouthaan, “Robust downlink beamforming in mul-tiuser MISO cognitive radio networks with imperfect channel-state information,” IEEETrans. Veh. Technol., vol. 59, no. 6, pp. 2852–2860, May 2010.

[36] A. Ghasemi and E. S. Sousa, “Asymptotic performance of collaborative spectrum sens-ing under correlated log-normal shadowing,” IEEE Commun. Lett., vol. 11, no. 1, pp.34–36, Jan. 2007.

[37] A. Ghasemi and E. S. Sousa, “Fundamental limits of spectrum sharing in fading envi-ronments,” IEEE Trans. Wireless Commun., vol. 6, no. 2, pp. 649–658, Feb. 2007.

[38] A. Goldsmith, S. A. Jafar, I. Maric, and S. Srinivasa, “Breaking spectrum gridlock withcognitive radios: An information theoretic perspective,” Proc. IEEE, vol. 97, no. 5, pp.894–914, May 2009.

[39] X. Gong and G. Ascheid, “Cooperative spectrum sensing over fading sensing andreporting channels,” in Proc. European Reconfigurable Radio Technol. Workshop (ERRT),Mainz, Germany, June 2010.

[40] X. Gong and G. Ascheid, “Ergodic capacity for cognitive radio with partial channelstate information of the primary user,” in Proc. IEEE Wireless Commun. and Netw. Conf.(WCNC), Paris, France, Apr. 2012.

[41] X. Gong, G. Dartmann, A. Ispas, and G. Ascheid, “Optimal power allocation in a spec-trum sharing system with partial CSI,” in Proc. IEEE Veh. Tech. Conf. (VTC-Spring), Yoko-hama, Japan, May 2012.

[42] X. Gong, A. Ishaque, G. Dartmann, and G. Ascheid, “MSE-based linear transceiveroptimization in MIMO cognitive radio networks with imperfect channel knowledge,”in Proc. IAPR Workshop Cognitive Inf. Process. (CIP), Elba, Italy, Jun. 2010.

158 BIBLIOGRAPHY

[43] X. Gong, A. Ishaque, G. Dartmann, and G. Ascheid, “Duality-based robust transceiverdesign for cognitive downlink systems,” in Proc. IEEE Veh. Technol. Conf. (VTC-Fall), SanFrancisco, CA, USA, Sep. 2011.

[44] X. Gong, A. Ishaque, G. Dartmann, and G. Ascheid, “MSE-based stochastic transceiveroptimization in downlink cognitive radio networks,” in Proc. IEEE Wireless Commun. andNetw. Conf. (WCNC), Cancun, Mexico, Mar. 2011.

[45] X. Gong, A. Ispas, and G. Ascheid, “GLRT-based cooperative sensing in cognitive radionetworks with partial CSI,” in Proc. Int. ICST Conf. Cognitive Radio Oriented Wireless Netw.(Crowncom), Osaka, Japan, June 2011.

[46] X. Gong, A. Ispas, and G. Ascheid, “GLRT-based cooperative spectrum sensing forcognitive radio with rank information,” in Proc. IEEE Workshop Stat. Signal Process. (SSP),Nice, France, June 2011.

[47] X. Gong, A. Ispas, and G. Ascheid, “Outage-constrained power control in spectrumsharing systems with partial primary CSI,” in Proc. IEEE Global Commun. Conf. (GLOBE-COM), Anaheim, CA, USA, Dec. 2012.

[48] X. Gong, A. Ispas, G. Dartmann, and G. Ascheid, “Power allocation and performanceanalysis in spectrum sharing systems with statistical CSI,” IEEE Trans. Wireless Commun.,vol. 12, no. 4, pp. 1819–1831, Apr. 2013.

[49] X. Gong, A. Ispas, G. Dartmann, and G. Ascheid, “Outage-constrained power allocationin spectrum sharing systems with partial CSI,” IEEE Trans. Commun., vol. 62, no. 2, pp.452–466, Feb. 2014.

[50] X. Gong, M. Jordan, A. Ishaque, G. Dartmann, and G. Ascheid, “Robust MSE-basedtransceiver optimization in MISO downlink cognitive radio network,” in Proc. IEEEWireless Commun. and Netw. Conf. (WCNC), Sydney, Australia, Apr. 2010.

[51] X. Gong, B. Prasse, A. Ispas, and G. Ascheid, “Sensing-based power allocation in cogni-tive radio networks with partial CSI,” 2014, preprint.

[52] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. SanDiego, CA, USA: Academic Press, 2004.

[53] C. Han, I. Nevat, and J. Yuan, “Spectrum sensing in cooperative cognitive radionetworks with partial CSI,” June 2010. http://arxiv.org/abs/1006.3154

[54] N. Hao and S. J. Yoo, “Short-range cognitive radio network: System architecture andMAC protocol for coexistence with legacy WLAN,” Wireless Netw., vol. 17, no. 2, pp.507–530, Feb. 2011.

[55] S. Haykin, “Cognitive radio: Brain-empowered wireless communications,” IEEE J. Sel.Areas Commun., vol. 23, no. 2, pp. 201–220, Feb. 2005.

[56] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge, UK: Cambridge UniversityPress, 1985.

[57] R. Hunger, M. Joham, and W. Utschick, “On the MSE-duality of the broadcast channeland the multiple access channel,” IEEE Trans. Signal Process., vol. 57, no. 2, pp. 698–713,Feb. 2009.

http://arxiv.org/abs/1006.3154

BIBLIOGRAPHY 159

[58] A. D. Ioffe and V. M. Tihomirov, Theory of Extremal Problems. Moscow, Russia: Nauka,1974, English version: North-Holland, Amsterdam, Netherlands, 1979.

[59] A. Ishaque, “Robust transceiver optimization in multi-antenna cognitive radio systems,”Master’s thesis, RWTH Aachen University, Aachen, Germany, 2010.

[60] M. H. Islam, Y. C. Liang, and A. T. Hoang, “Joint power control and beamforming forcognitive radio networks,” IEEE Trans. Wireless Commun., vol. 7, no. 7, pp. 2415–2419,Jul. 2008.

[61] P. Jia, M. Vu, T. Le-Ngoc, S. C. Hong, and V. Tarokh, “Capacity- and bayesian-basedcognitive sensing with location side information,” IEEE J. Sel. Areas Commun., vol. 29,no. 2, pp. 276–289, Feb. 2011.

[62] N. Jindal, S. Vishwanath, and A. Goldsmith, “On the duality of Gaussian multiple-accessand broadcast channels,” IEEE Trans. Inf. Theory, vol. 50, no. 5, pp. 768–783, May 2004.

[63] S. Kandukuri and S. Boyd, “Optimal power control in interference-limited fading wire-less channels with outage-probability specifications,” IEEE Trans. Wireless Commun.,vol. 1, no. 1, pp. 46–55, Jan. 2002.

[64] X. Kang, Y. C. Liang, and A. Nallanathan, “Optimal power allocation for fading channelsin cognitive radio networks under transmit and interference power constraints,” in Proc.IEEE Int. Conf. Commun. (ICC), Beijing, China, May 2008, pp. 3568–3572.

[65] X. Kang, R. Zhang, Y. C. Liang, and H. K. Garg, “Optimal power allocation strategiesfor fading cognitive radio channels with primary user outage constraint,” IEEE J. Sel.Areas Commun., vol. 29, no. 2, pp. 374–383, Feb. 2011.

[66] S. M. Kay, Fundamentals of Statistical Signal Processing: Detection Theory. EnglewoodCliffs, NJ, USA: Prentice Hall, 1998, vol. 2.

[67] C. T. Kelley, Solving Nonlinear Equations with Newton’s Method. Philadephia, PA, USA:Society for Industrial and Applied Mathematics (SIAM), 2003.

[68] J. P. Kermoal, L. Schumacher, K. I. Pedersen, P. E. Mogensen, and F. Frederiksen, “Astochastic MIMO radio channel model with experimental validation,” IEEE J. Sel. AreasCommun., vol. 20, no. 6, pp. 1211–1226, Aug. 2002.

[69] M. A. Khojastepour and B. Aazhang, “The capacity of average and peak power con-strained fading channels with channel side information,” in Proc. IEEE Wireless Commun.and Netw. Conf. (WCNC), Atlanta, USA, Mar. 2004.

[70] M. G. Khoshkholgh, K. Navaie, and H. Yanikomeroglu, “Access strategies for spectrumsharing in fading environment: Overlay, underlay, and mixed,” IEEE Trans. Mobile Com-put., vol. 9, no. 12, pp. 1780–1793, Dec. 2010.

[71] M. Khoshnevisan and J. Laneman, “Power allocation in multi-antenna wireless systemssubject to simultaneous power constraints,” IEEE Trans. Commun., vol. 60, no. 12, pp.3855–3864, Dec. 2012.

[72] H. V. Khuong and H. Y. Kong, “General expression for PDF of a sum of independentexponential random variables,” IEEE Commun. Lett., vol. 10, no. 3, pp. 159–161, Mar.2006.

160 BIBLIOGRAPHY

[73] H. Kim, H. Wang, S. Lim, and D. Hong, “On the impact of outdated channel informa-tion on the capacity of secondary user in spectrum sharing environments,” IEEE Trans.Wireless Commun., vol. 11, no. 1, pp. 284–295, Jan. 2012.

[74] D. E. Kirk, Optimal Control Theory: An Introduction. Mineola, NY, USA: Dover Publica-tions, 2004.

[75] S. Kritchman and B. Nadler, “Non-parametric detection of the number of signals: Hy-pothesis testing and random matrix theory,” IEEE Trans. Signal Process., vol. 57, no. 10,pp. 3930–3941, Oct. 2009.

[76] E. L. Lehmann and J. P. Romano, Testing Statistical Hypotheses. New York, USA: JohnWiley and Sons, 1959.

[77] K. B. Letaief and W. Zhang, “Cooperative communications for cognitive radio net-works,” Proc. IEEE, vol. 97, no. 5, pp. 878–893, May. 2009.

[78] Z. Li, F. R. Yu, and M. Huang, “A distributed consensus-based cooperative spectrum-sensing scheme in cognitive radios,” IEEE Trans. Veh. Technol., vol. 59, no. 1, pp. 383–393,Jan. 2010.

[79] Y. C. Liang, Y. Zeng, E. Peh, and A. T. Hoang, “Sensing-throughput tradeoff for cognitiveradio networks,” in Proc. IEEE Int. Commun. Conf. (ICC), Glasgow, Scotland, June 2007,pp. 5330–5335.

[80] S. Lim, H. Wang, H. Kim, and D. Hong, “Mean value-based power allocation without in-stantaneous CSI feedback in spectrum sharing systems,” IEEE Trans. Wireless Commun.,vol. 11, no. 3, pp. 874–879, Mar. 2012.

[81] A. Limmanee, S. Dey, and J. S. Evans, “Service-outage capacity maximization in cogni-tive radio for parallel fading channels,” IEEE Trans. Commun., vol. 61, no. 2, pp. 507–520,Feb. 2013.

[82] J. Löberg, “Yalmip : A toolbox for modeling and optimization in MATLAB,” in Proc.the CACSD Conf., Taipei, Taiwan, 2004. http://control.ee.ethz.ch/~joloef/yalmip.php.

[83] M. S. Lobo, L. Vandenberghe, S. Boyd, and H. Lebret, “Applications of second-ordercone programming,” Linear Algebra and its Applications, vol. 284, no. 1-3, pp. 193–228,Nov 1998.

[84] R. López-Valcarce, G. Vazquez-Vilar, and J. Sala, “Multiantenna spectrum sensing forcognitive radio: Overcoming noise uncertainty,” in Proc. IAPR Workshop Cognitive Inf.Process. (CIP), Elba, Italy, Jun. 2010.

[85] D. G. Luenberger, Optimization by Vector Space Methods. New York, USA: John Wileyand Sons, 1968.

[86] J. Lunden, V. Koivunen, A. Huttunen, and H. V. Poor, “Collaborative cyclostationaryspectrum sensing for cognitive radio systems,” IEEE Trans. Signal Process., vol. 57, no. 11,pp. 4182–4195, Nov. 2009.

[87] J. Luo, R. Yates, and P. Spasojevic, “Service outage based power and rate allocation forparallel fading channels,” IEEE Trans. Inf. Theory, vol. 51, no. 7, pp. 2594–2611, July 2005.

http://control.ee.ethz.ch/~joloef/yalmip.php.

BIBLIOGRAPHY 161

[88] Z. Luo, W. Ma, A. M. C. So, Y. Ye, and S. Zhang, “Semidefinite relaxation of quadraticoptimization problems,” IEEE Signal Process. Mag., vol. 27, no. 3, pp. 20–34, May 2010.

[89] B. Makki and T. Eriksson, “On the capacity of Rayleigh-fading correlated spectrumsharing networks,” EURASIP J. Wireless Commun. and Netw., vol. 2011, no. 1, p. 83, Aug.2011.

[90] J. Mitola and G. Q. M. Jr., “Cognitive radio: Making software radios more personal,”IEEE Personal Commun., vol. 6, no. 4, pp. 13–18, Aug. 1999.

[91] H. Mu and J. K. Tugnait, “Joint soft-decision cooperative spectrum sensing and powercontrol in multiband cognitive radios,” IEEE Trans. Signal Process., vol. 60, no. 10, pp.5334–5346, Oct. 2012.

[92] R. J. Muirhead, Aspects of Multivariate Statistical Theory. New York, USA: John Wileyand Sons, 1982.

[93] L. Musavian and S. Aissa, “Outage-constrained capacity of spectrum-sharing channelsin fading environments,” IET Commun., vol. 2, no. 6, pp. 724–732, July 2008.

[94] L. Musavian and S. Aissa, “Fundamental capacity limits of cognitive radio in fadingenvironments with imperfect channel information,” IEEE Trans. Commun., vol. 57, no. 11,pp. 3472–3480, Nov. 2009.

[95] L. Musavian, M. R. Nakhi, M. Dohler, and A. H. Aghvami, “Effect of channel uncertaintyon the mutual information of MIMO fading channels,” IEEE Trans. Veh. Technol., vol. 56,no. 5, pp. 2798–2806, Sep. 2007.

[96] N. Jindal, “MIMO broadcast channels with finite-rate feedback,” IEEE Trans. Inf. Theory,vol. 52, no. 11, pp. 5045–5060, Nov. 2006.

[97] J. Oh and W. Choi, “A hybrid cognitive radio system: A combination of underlay andoverlay approaches,” in Proc. IEEE Veh. Technol. Conf. (VTC-Fall), Ottawa, Canada, Sep.2010.

[98] D. Palomar and Y. Jiang, “MIMO transceiver design via majorization theory,” Founda-tions and Trends in Communications and Information Theory, vol. 3, no. 4-5, pp. 331–551,2006.

[99] D. P. Palomar, “A unified framework for communications through MIMO channels,”Dissertation, Technical University of Catalonia, Barcelona, Spain, May 2003.

[100] J. S. Pang and G. Scutari, “Joint sensing and power allocation in nonconvex cognitiveradio games: Quasi-Nash equilibria,” IEEE Trans. Signal Process., 2013, accepted.http://arxiv.org/abs/1212.6225

[101] E. C. Y. Peh, Y. C. Liang, Y. Guan, and Y. Zeng, “Optimization of cooperative sens-ing in cognitive radio networks: A sensing-throughput tradeoff view,” IEEE Trans. Veh.Technol., vol. 58, no. 9, pp. 5294–5299, Nov. 2009.

[102] E. C. Y. Peh, Y. C. Liang, Y. Guan, and Y. Zeng, “Power control in cognitive radios un-der cooperative and non-cooperative spectrum sensing,” IEEE Trans. Wireless Commun.,vol. 10, no. 12, pp. 4238–4248, Dec. 2011.

http://arxiv.org/abs/1212.6225

162 BIBLIOGRAPHY

[103] H. V. Poor, An introduction to signal detection and estimation. New York, NY, USA:Springer-Verlag, 1994.

[104] B. Prasse, “Interference mitigation in hybrid-cognitive-radio systems,” Bachelor Thesis,RWTH Aachen University, 2012.

[105] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: TheArt of Scientific Computing. Cambridge, UK: Cambridge University Press, 2007.

[106] Z. Quan, S. Cui, and A. H. Sayed, “Optimal linear cooperation for spectrum sensing incognitive radio networks,” IEEE J. Sel. Topics Signal Process., vol. 2, no. 1, pp. 28–40, Feb.2008.

[107] Z. Quan, S. Cui, A. H. Sayed, and H. V. Poor, “Optimal multiband joint detection forspectrum sensing in cognitive radio networks,” IEEE Trans. Signal Process., vol. 57, no. 3,pp. 1128–1140, Mar. 2009.

[108] D. Ramirez, G. Vazquez-Vilar, R. Lopez-Valcarce, J. Via, and I. Santamaria, “Detection ofrank-P signals in cognitive radio networks with uncalibrated multiple antennas,” IEEETrans. Signal Process., vol. 59, no. 8, pp. 3764–3774, Aug. 2011.

[109] T. Rappaport, Wireless Communications: Principles and Practice, 2nd ed. Upper SaddleRiver, NJ, USA: Prentice Hall, 2001.

[110] Z. Rezki and M. S. Alouini, “Ergodic capacity of cognitive radio under imperfect channelstate information,” IEEE Trans. Veh. Technol., vol. 61, no. 5, pp. 2108–2119, June 2012.

[111] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, UK: Cambridge Uni-versity Press, 2004.

[112] S. Boyd and L. Vandenberghe, “Subgradients,” in Notes for EE364b, Stanford University,Winter 2006-07, Apr. 2008.

[113] A. Sahai and D. Cabric, “Spectrum sensing: Fundamental limits and practical chal-lenges,” in Proc. IEEE Int. Symp. New Frontiers Dynamic Spectrum Access Networks (DyS-PAN), Baltimore, MD, USA, Nov. 2005.

[114] S. Shi, M. Schubert, and H. Boche, “Downlink MMSE transceiver optimization for mul-tiuser MIMO systems: MMSE balancing,” IEEE Trans. Signal Process., vol. 56, no. 8, pp.3702–3712, Aug. 2007.

[115] L. V. Snyder and Z.-J. M. Shen, Fundamentals of Supply Chain Theory. Hoboken, NJ, USA:John Wiley and Sons, 2011.

[116] K. Son, B. C. Jung, S. Chong, and D. K. Sung, “Power allocation policies with full andpartial inter-system channel state information for cognitive radio networks,” WirelessNetw., vol. 19, no. 1, pp. 99–113, Jan. 2013.

[117] S. Srinivasa and S. A. Jafar, “Soft sensing and optimal power control for cognitive radio,”IEEE Trans. Wireless Commun., vol. 9, no. 12, pp. 3638–3649, Dec. 2010.

[118] C. R. Stevenson, G. Ghouinard, Z. Lei, W. Hu, S. J. Shellhammer, and W. Caldwell,“IEEE 802.22: The first cognitive radio wireless regional area network standard,” IEEECommun. Mag., vol. 47, no. 1, pp. 130–138, Jan. 2009.

BIBLIOGRAPHY 163

[119] S. Stotas and A. Nallanathan, “Enhancing the capacity of spectrum sharing cognitiveradio networks,” IEEE Trans. Veh. Technol., vol. 60, no. 8, pp. 3768–3779, Oct. 2011.

[120] S. Stotas and A. Nallanathan, “On the outage capacity of sensing-enhanced spectrumsharing cognitive radio systems in fading channels,” IEEE Trans. Commun., vol. 59,no. 10, pp. 2871–2882, Oct. 2011.

[121] H. A. Suraweera, P. J. Smith, and M. Shafi, “Capacity limits and performance analysisof cognitive radio with imperfect channel knowledge,” IEEE Trans. Veh. Technol., vol. 59,no. 4, pp. 1811–1822, May 2010.

[122] A. Taherpour, M. Nasiri-Kenari, and S. Gazor, “Multiple antenna spectrum sensing incognitive radios,” IEEE Trans. Wireless Commun., vol. 9, no. 2, pp. 814–823, Feb. 2010.

[123] R. Tandra and A. Sahai, “SNR walls for feature detectors,” in Proc. IEEE Int. Symp. NewFrontiers Dynamic Spectrum Access Networks (DySPAN), Apr. 2007, pp. 559–570.

[124] R. Tandra and A. Sahai, “SNR walls for signal detection,” IEEE J. Sel. Topics Signal Pro-cess., vol. 2, no. 1, pp. 4–17, Feb 2008.

[125] R. Tandra, A. Sahai, and S. M. Mishra, “What is a spectrum hole and what does it taketo recognize one?” Proc. IEEE, vol. 97, no. 5, pp. 824–848, May 2009.

[126] G. Taricco and E. Riegler, “On the ergodic capacity of correlated Rician fading MIMOchannels with interference,” IEEE Trans. Inf. Theory, vol. 57, no. 7, pp. 4123–4137, July2011.

[127] T. Terlaky and J. Zhu, “Comments on "Dual methods for nonconvex spectrum optimiza-tion of multicarrier systems",” Optimization Lett., vol. 2, no. 4, pp. 497–503, 2008.

[128] A. Tkachenko, A. D. Cabric, and R. W. Brodersen, “Cyclostationary feature detectorexperiments using reconfigurable BEE2,” in Proc. IEEE Int. Symp. New Frontiers DynamicSpectrum Access Networks (DySPAN), Apr. 2007, pp. 216–219.

[129] H. Urkowitz, “Energy detection of unknown deterministic signals,” Proc. IEEE, vol. 55,no. 4, pp. 523–531, Apr. 1967.

[130] M. Vu, N. Devroye, and V. Tarokh, “On the primary exclusive region of cognitive net-works,” IEEE Trans. Wireless Commun., vol. 8, no. 7, pp. 3380–3385, July 2009.

[131] N. Vucic, H. Boche, and S. Shi, “Robust transceiver optimization in downlink multiuserMIMO systems,” IEEE Trans. Signal Process., vol. 57, no. 9, pp. 3576–3587, Sep. 2009.

[132] P. Wang, J. Fang, N. Han, and H. Li, “Multiantenna-assisted spectrum sensing for cog-nitive radio,” IEEE Trans. Veh. Technol., vol. 59, no. 4, pp. 1791–1800, May. 2010.

[133] M. Wax and T. Kailath, “Detection of signals by information theoretic criteria,” IEEETran. on Acoustics, Speech, and Signal Process., vol. 33, no. 2, pp. 387–392, Apr. 1985.

[134] M. Wellens, A. de Baynast, and P. Mahonen, “Performance of dynamic spectrum accessbased on spectrum occupancy statistics,” IET Commun., vol. 2, no. 6, pp. 772–782, July2008.

[135] A. Wiesel, Y. Eldar, and S. Shamai, “Linear precoding via conic optimization for fixedMIMO receivers,” IEEE Trans. Signal Process., vol. 54, no. 1, pp. 161–176, Jan. 2006.

164 BIBLIOGRAPHY

[136] X. Yang, S. Peng, K. Lei, R. Lu, and X. Cao, “Impact of the dimension of the observationspace on the decision thresholds for GLRT detectors in spectrum sensing,” IEEE WirelessCommun. Lett., vol. 1, no. 4, pp. 396–399, Aug. 2012.

[137] S. Yarkan and H. Arslan, “Exploiting location awareness toward improved wireless sys-tem design in cognitive radio,” IEEE Commun. Mag., vol. 46, no. 1, pp. 128–136, Jan.2008.

[138] W. Yu and R. Lui, “Dual methods for nonconvex spectrum optimization of multicarriersystems,” IEEE Trans. Commun., vol. 54, no. 7, pp. 1310–1322, Jul. 2006.

[139] T. Yucek and H. Arslan, “A survey of spectrum sensing algorithms for cognitive radioapplications,” IEEE Commun. Surveys Tutorials, vol. 11, no. 1, pp. 116–130, First quarter2009.

[140] Y. Zeng and Y. C. Liang, “Eigenvalue-based spectrum sensing algorithms for cognitiveradio,” IEEE Trans. Commun., vol. 57, pp. 1784–1793, June 2009.

[141] Y. Zeng and Y. C. Liang, “Spectrum-sensing algorithms for cognitive radio based onstatistical covariances,” IEEE Trans. Veh. Technol., vol. 58, pp. 1804–1815, May 2009.

[142] Y. Zeng, Y. C. Liang, A. T. Hoang, and R. Zhang, “A review on spectrum sensing forcognitive radio: Challenges and solutions,” EURASIP J. Advances in Signal Process., 2010,article ID 381465.

[143] L. Zhang, Y. C. Liang, and Y. Xin, “Joint beamforming and power allocation for multipleaccess channels in cognitive radio networks,” IEEE J. Sel. Areas Commun., vol. 26, no. 1,pp. 38–51, Jan. 2008.

[144] L. Zhang, Y. C. Liang, Y. Xin, and H. V. Poor, “Robust cognitive beamforming withpartial channel state information,” IEEE Trans. Wireless Commun., vol. 8, no. 8, pp. 4143–4153, Aug. 2009.

[145] L. Zhang, R. Zhang, Y. C. Liang, Y. Xin, and H. V. Poor, “On Gaussian MIMO BC-MACduality with multiple transmit covariance constraints,” IEEE Trans. Inf. Theory, vol. 58,no. 4, pp. 2064–2078, Apr. 2012.

[146] R. Zhang, “On peak versus average interference power constraints for protecting pri-mary users in cognitive radio networks,” IEEE Trans. Wireless Commun., vol. 8, no. 4, pp.2112–2120, Apr.

[147] R. Zhang, Y. C. Liang, and S. Cui, “Dynamic resource allocation in cognitive radionetworks: A convex optimization perspective,” IEEE Signal Process. Mag., vol. 27, no. 3,pp. 102–114, May 2010.

[148] R. Zhang, T. J. Lim, Y. C. Liang, and Y. Zeng, “Multi-antenna based spectrum sensingfor cognitive radios: A GLRT approach,” IEEE Trans. Commun., vol. 58, no. 1, pp. 84–88,Jan. 2010.

[149] R. Zhang, “Optimal power control over fading cognitive radio channel by exploitingprimary user CSI,” in Proc. IEEE Global Commun. Conf. (GLOBECOM), New Orleans,LA, USA, Dec. 2008.

BIBLIOGRAPHY 165

[150] W. Zhang, R. Mallik, and K. Letaief, “Optimization of cooperative spectrum sensingwith energy detection in cognitive radio networks,” IEEE Trans. Wireless Commun., vol. 8,no. 12, pp. 5761–5766, Dec. 2009.

[151] X. Zhang, D. Palomar, and B. Ottersten, “Statistically robust design of linear MIMOtransceivers,” IEEE Trans. Signal Process., vol. 56, no. 8, pp. 3678–3689, July 2008.

[152] Q. Zhao and B. Sadler, “A survey of dynamic spectrum access,” IEEE Signal Process.Mag., vol. 24, no. 3, pp. 79–89, May 2007.

[153] G. Zheng, K. K. Wong, and B. Ottersten, “Robust cognitive beamforming with boundedchannel uncertainties,” IEEE Trans. Signal Process., vol. 57, no. 12, pp. 4871–4881, Dec.2009.

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